Thermodynamics Practice Problems

Decide which temperature the question wants

Almost every avoidable error in thermal physics is a temperature error, and there are only two cases to tell apart. A gas law divides by absolute temperature, so it needs kelvin and nothing else will do. Specific heat and thermal expansion take a temperature difference, and a rise of 10 °C is a rise of 10 K, so either scale gives the same number there.

Work out which of the two you are in before touching a calculator. If the symbol is T on its own, convert. If it is ΔT, do not, because subtracting two Celsius temperatures has already removed the offset. Getting this wrong on a gas law is not a small error: 20 °C to 40 °C looks like a doubling but is 293.15 K to 313.15 K, a rise of 6.8 percent, so the Celsius version overstates the increase about fifteenfold.

Know the constants that carry a hidden mass unit

Specific heat capacity and specific latent heat are both quoted per gram or per kilogram, and the two differ by a thousand. Water is 4.184 J/(g·K) or 4184 J/(kg·K), and its latent heat of fusion is 334 J/g or 334,000 J/kg. Pairing a per-kilogram constant with a mass in grams is the classic factor-of-1000 slip, and the answer it produces still looks like an energy.

The questions here quote whichever unit the matching calculator displays, so read the unit on the value rather than assuming. A quick sanity anchor helps: heating a mug of water, about 250 g, by 80 K takes roughly 84 kJ, and a 2 kW kettle therefore needs about 42 seconds. Anything three orders of magnitude away from that kind of anchor is a unit problem, not a physics problem.

Check the direction before the digits

Every relationship in this topic has an obvious direction, and checking it catches inverted fractions instantly. Compress a gas and its volume falls. Warm it at constant pressure and the volume rises. Put more heat into a fixed mass and the temperature rise is larger. Give the same heat to a larger mass and the rise is smaller.

For the combined gas law there is a stronger check available: PV divided by T is the same before and after. If the two sides disagree, an input is wrong, and the most likely culprit is a temperature left in Celsius or a pressure quoted in different units in the two states. The units only have to match each other, since the equation is a ratio, but they do have to match.

Where thermodynamics answers go wrong

• Celsius in a gas law. The equation divides by T, so an arbitrary zero makes it meaningless and below freezing it returns a negative volume.
• Mixing per-gram and per-kilogram constants. A clean factor of 1000, and the result still has the units of energy, so nothing looks wrong.
• Using Q = mcΔT across a phase change. Melting and boiling happen at constant temperature, so they need Q = mL. A problem spanning both needs separate stages, and the phase changes usually dominate the total.
• Dropping the 10⁻⁶ on an expansion coefficient. Tables print steel as 12, meaning 12 × 10⁻⁶ per kelvin. Using 12 directly overstates the expansion a million times.
• Assuming volume ratios follow temperature ratios in Celsius. They follow absolute temperature, which is why a 20 degree rise near room temperature is a few percent rather than a doubling.
• Treating a changing amount of gas as a combined-law problem. The law only holds for a sealed sample, because that is what lets the amount of gas cancel from both sides.