Topic 9.5: Specific Heat and Thermal Conductivity

What Topic 9.5 requires

Topic 9.5 sits in Unit 9, Thermodynamics, weighted at 15 to 18 percent of the multiple-choice section with a suggested 10 to 16 class periods. It carries two learning objectives, four essential knowledge statements, and one boundary statement.

9.5.A, describe the energy required to change the temperature of an object by a certain amount.

• 9.5.A.1 states that the amount of energy required to change the temperature of a material is related to the material's specific heat, and prints the relevant equation $Q = mc\Delta T$.
• 9.5.A.2 states that the specific heat of a material is an intrinsic property of that material that depends on the arrangement and interactions of the atoms that make up the material.

9.5.B, describe the rate at which energy is transferred by conduction through a given material.

• 9.5.B.1 states that the rate at which energy is transferred by conduction through a given material is related to the thermal conductivity, the physical dimensions of the material, and the temperature difference across the material, and prints the relevant equation $\dfrac{Q}{\Delta t} = \dfrac{kA\Delta T}{L}$.
• 9.5.B.2 states that the thermal conductivity of a material is an intrinsic property of that material that depends on the arrangement and interactions of the atoms that make up the material.

Boundary statement, in full: "AP Physics 2 will model specific heat as independent of temperature." That is the entire boundary statement, one sentence. Note that 9.5.A.2 and 9.5.B.2 are the same sentence with one noun swapped: $c$ and $k$ are both intrinsic properties of the material.

The CED lists five suggested skills here, more than any Unit 9 topic except 9.2: 1.B, create quantitative graphs with appropriate scales and units, including plotting data; 2.B, calculate or estimate an unknown quantity with units from known quantities, by selecting and following a logical computational pathway; 2.D, predict new values or factors of change of physical quantities using functional dependence between variables; 3.A, create experimental procedures that are appropriate for a given scientific question; and 3.B, apply an appropriate law, definition, theoretical relationship, or model to make a claim.

Topic 9.5 is the only one of Unit 9's six topics with 2.B listed, a fair signal that this is the unit's calculation topic. One caveat the CED adds is worth carrying: Progress Check questions are built on that content-and-skill pairing, but "AP Exam questions can pair the content with any of the skills." Treat the list as emphasis, not a fence.

Q = mc(deltaT), symbol by symbol (9.5.A.1)

Every symbol is defined by the AP Physics 2 equation sheet itself: the Thermal Physics column's symbol list reads $Q$ as "energy transferred to a system by heating", $m$ as mass, $c$ as specific heat, and $T$ as temperature. Three things follow.

$Q$ carries a sign, and the sheet's wording fixes which one. Energy transferred to the system is positive. Warm something up and $\Delta T > 0$, so $Q > 0$. That is the same convention as $\Delta U = Q + W$ in Topic 9.4, and it is the same $Q$: the first law's heating term, priced for a particular object.

The units of $c$ come out of the equation. Rearranged, $c = \dfrac{Q}{m\Delta T}$, so specific heat is measured in joules per kilogram per kelvin, $\mathrm{J/(kg \cdot K)}$. Read that off the rearrangement rather than memorizing it.

$\Delta T$ is a difference, so kelvin and degrees Celsius give the same number. The two intervals are the same size, so a change of 25 K and a change of 25 degrees Celsius are the same change. This is the one place in Unit 9 where the distinction does not bite. Wherever an absolute temperature appears, including $PV = nRT$ in Topic 9.2, $K_{\text{avg}} = \frac{3}{2}k_B T$ in Topic 9.1, and $U = \frac{3}{2}nRT$ in Topic 9.4, it must be in kelvin.

Solve for the temperature change instead and the comparison questions answer themselves:

Give two objects the same $Q$ and the one with the larger $mc$ warms less. That product is the object's energy cost per kelvin, and it decides a calorimetry problem. The CED never names it, and the phrase "heat capacity" appears nowhere in the AP Physics 2 course description, so write it as $mc$ and keep "specific heat" for $c$ alone.

Specific heat is a property of the material, not of the sample (9.5.A.2)

Essential knowledge 9.5.A.2 calls specific heat an intrinsic property, and that word does precise work. An intrinsic property does not change when you take more or less of the substance: cut a copper block in half and each half has the same specific heat as the whole, just as each half has the same density. What changes is $Q$, because $m$ halved. A question that hands you a specific heat is handing you a fact about a substance, and the number you compute from one experiment is the one you use in the next.

The rest of 9.5.A.2 says where the property comes from: the arrangement and interactions of the atoms that make up the material. The CED stops there and builds no microscopic model, so neither should an answer. The nearest the course comes is essential knowledge 9.4.A.1, which makes internal energy the sum of the kinetic energy of the objects in a system and the potential energy of their configuration. Energy you add can land in either place, and temperature tracks only the motion. No question will ask you to compute a specific heat from atomic structure.

The consequence you are asked about directly is the comparison: two 1.0 kg samples given 5000 J each end at different temperatures, and the one with the larger $c$ moves less.

Neither the equation sheet nor the CED prints a table of specific heat values. Any question that needs one supplies it, exactly as the CED's own sample multiple-choice set does when it gives the specific heat of water as $4184 \, \mathrm{J/(kg \cdot K)}$ inside the question stem. Do not carry remembered values into an exam, and do not treat a missing $c$ as something to look up: if it is not in the stem, the question wants you to solve for it.

The boundary statement: specific heat is modeled as independent of temperature

"AP Physics 2 will model specific heat as independent of temperature."

Short, and load-bearing. What it grants you:

• $c$ is a constant you can pull out of any calculation. You never have to ask what the specific heat was at 290 K versus at 350 K.
• $Q = mc\Delta T$ is exact within the course model, not an approximation over a small range. A 200 K change takes the same one-line calculation as a 2 K one.
• The graphs are straight. Plot $Q$ against $\Delta T$ for a fixed sample and you get a line through the origin with slope $mc$. Skill 1.B lives on that linearity, and the boundary statement is what guarantees it.

What it does not say: it does not say specific heat is the same for every material, because 9.5.A.2 says the opposite; it does not say $c$ is independent of anything other than temperature; and it does not say a substance has the same specific heat in every phase.

On phases: there is no latent-heat or phase-change equation anywhere on the AP Physics 2 equation sheet, and Unit 9's required course content contains no phase-change essential knowledge statement. The Thermal Physics column holds exactly eight equations, and none of them describes melting or boiling.

The conduction rate, Q over delta t equals kA delta T over L (9.5.B.1)

The left-hand side is not an energy, it is a rate: joules per second, which is watts. That makes it the same kind of quantity as power, and the same sheet prints $P_{\text{avg}} = \dfrac{\Delta E}{\Delta t}$ in its mechanics column. If your answer to a conduction question came out in joules, you answered a different question.

Essential knowledge 9.5.B.1 says the rate is related to "the thermal conductivity, the physical dimensions of the material, and the temperature difference across the material". The CED does not spell out which dimension goes where. The equation does, and swapping them is the standard error.

• $A$ is the area of the face the energy crosses, the broad side of the slab.
• $L$ is the length of the path through the material, measured along the direction the energy travels: the thickness of the window, not its width.
• $\Delta T$ is the temperature difference across the material, between the two faces. Another difference, so either scale works.
• $k$ is the thermal conductivity, and its units follow from the equation: watts per meter per kelvin, $\mathrm{W/(m \cdot K)}$.

Every dependence is first order, which makes this a natural home for skill 2.D. Hold everything else fixed and:

One reading questions probe: the equation contains a single $\Delta T$, so it gives the rate for that temperature difference. If the two sides are free to equilibrate, $\Delta T$ shrinks as energy moves and the rate shrinks with it, so multiplying an initial rate by a long time overestimates the energy transferred.

Topic 9.3 supplies the context: essential knowledge 9.3.A.2 names conduction, convection, and radiation as the three thermal processes by which energy may be transferred between systems at different temperatures. Topic 9.5 puts a rate equation on exactly one of them. The sheet does print $P = A\sigma T^4$ for radiated power, but in the Modern Physics column, and no Unit 9 essential knowledge statement uses it. Convection gets no equation at all.

The k problem: one letter, three quantities, one equation sheet

Say this once and then never be caught by it. The AP Physics 2 equation sheet uses the letter $k$ for three unrelated quantities, and its own symbol lists are where you find out which is which.

Two near misses sit alongside them. $k_B$, with the subscript, is Boltzmann's constant, $1.38 \times 10^{-23} \, \mathrm{J/K}$, and it appears in $PV = Nk_B T$ and $K_{\text{avg}} = \frac{3}{2}k_B T$ in this same unit. And $\kappa$, lowercase kappa, in the Electricity column is the dielectric constant, not a $k$ at all despite looking like one in handwriting.

The dangerous one is the Coulomb constant, for a specific reason: it is the only $k$ on the sheet with a number printed next to it, so a student who has drilled "$k$ is $9.0 \times 10^9$" can walk that value straight into a conduction problem. Thermal conductivity has no sheet value and never will, because 9.5.B.2 makes it an intrinsic property: it differs for glass and for copper, and a question that needs it states it. If you are reaching for a $k$ the problem did not supply, you are in the wrong column.

Thermal conductivity is intrinsic too (9.5.B.2), and why the tile floor feels colder

The Unit 9 opener lists four essential questions, which the CED calls thought-provoking questions that motivate students and inspire inquiry. The first is: "Why does the tile floor in the bathroom feel so much colder than the bathroom mat?" It is Topic 9.5's question, and the two essential knowledge statements above answer it completely.

Start with what is not different. The tile and the mat have been in the same room all night. By essential knowledge 9.3.A.4, thermal equilibrium results when no net energy is transferred by thermal processes between two systems in thermal contact, and that is what the room has settled into. A thermometer reads the same on both. The tile is not colder.

What differs is $k$. Your foot is warmer than the floor, so energy conducts out of it into whatever you stand on, at the rate $\dfrac{kA\Delta T}{L}$. The contact area and the temperature difference are about the same either way, so the rate scales with the thermal conductivity, and tile conducts far better than a fabric mat. "Feels cold" reports the rate at which your skin is losing energy, not the temperature of the surface. It is why a metal railing and a wooden one on the same porch feel so different on a winter morning.

Essential knowledge 9.5.B.2 supplies the same sentence the CED used for specific heat: thermal conductivity is an intrinsic property of the material, set by the arrangement and interactions of its atoms. Neither property depends on how much material you have, and both are things the exam gives you rather than things you memorize.

One caution, because the course does not support the shortcut. Specific heat and thermal conductivity are independent: a large $c$ does not imply a large $k$. When a question asks which of two objects warms faster, decide first whether it is about the energy required, which is $c$, or the rate of supply, which is $k$.

Measuring c and k: experimental design and the graphs (skills 3.A, 1.B)

Skill 3.A, create experimental procedures appropriate for a given scientific question, appears on exactly two of Unit 9's six topic pages: 9.2 and 9.5. Skill 1.B, create quantitative graphs with appropriate scales and units, appears on the same two.

Two standard designs measure a specific heat. Steady heating: warm a known mass with a heater of known power $P$ and plot temperature against time, giving $c = P/(m \times \text{slope})$, as in the third worked example. That line is straight because of the boundary statement, not because the data were kind. Mixing: bring a hot sample into an insulated container with a known mass of water, wait for equilibrium, and solve $Q_{\text{hot}} + Q_{\text{cold}} = 0$, as in the first. Every choice there is aimed at making that zero true, which is what the insulation is for.

For a thermal conductivity, hold a slab between a source and a sink at a fixed temperature difference and measure the rate at which energy crosses it. The 1.B move is to choose axes that linearize the relationship before you plot:

Plotting rate against $L$ itself gives a curve you cannot read a slope off; against $1/L$ it is a line through the origin. That linearizing step is the graded part of a 1.B question more often than the plotting is.

Every one of these designs assumes all the energy you supply goes where you think it goes. If some leaks away, the temperature rises more slowly than it should and a specific heat calculated as if nothing leaked comes out too large.

How Topic 9.5 is tested, and where it leads

The routines are narrow enough to list, and rehearsing them is most of the preparation.

1. One object, one equation. Find $Q$, $m$, $c$, or $\Delta T$ from $Q = mc\Delta T$ (skill 2.B). Watch the sign, and watch that $\Delta T$ is a difference and not a temperature.
2. Two objects reaching a common temperature. An insulated container, a hot thing and a cold thing, and $Q_{\text{hot}} + Q_{\text{cold}} = 0$. Solve for an unknown specific heat, mass, or final temperature (skill 2.B).
3. Conduction rate through a slab (skill 2.B), or a comparison of two slabs differing in one variable.
4. Factor-of-change questions on either equation (skill 2.D). Both are products and quotients of first powers, so set them up as ratios rather than recomputing.
5. Design or critique an experiment to measure $c$ or $k$ (skill 3.A), including saying what the insulation is for.
6. Justify a claim with one of the two relationships as evidence (skill 3.B). The tile-floor question is the archetype.

The CED's sample multiple-choice set is fifteen questions long, and its answer key aligns question 9 to learning objective 9.5.A, essential knowledge 9.5.A.1, and skill 2.B. That question drops a hot metal sample into water in an insulated container and asks for the metal's specific heat: routine 2 above, and the shape of the first worked example below. Those questions illustrate the format of the exam, so read the alignment as a shape worth practising, not as a claim about how often anything appears.

The unit opener also flags the fourth free-response question, the Qualitative/Quantitative Translation, which "first requires students to make a claim and provide evidence and reasoning to support their claim without reference to equations" before asking for an equation and then for the connection between the two. Rehearse the verbal form of both relationships: more energy for a bigger temperature change, faster conduction through a thinner slab.

Where the topic sits:

• Topic 9.3 says energy moves spontaneously from hot to cold, that conduction is one of three ways it moves, and that equilibrium is where it stops. Topic 9.5 puts numbers on two of those.
• Topic 9.4 supplies $\Delta U = Q + W$, and its $Q$ is this $Q$. The PV diagrams guide walks that bookkeeping end to end.
• Topic 9.6 takes the equilibrium temperature a calorimetry calculation produces and asks what the first law cannot: why the process runs that way and not the other.
• The full sheet is at the AP Physics 2 formula sheet, the course overview at AP Physics 2.