What is the energy stored “inside” a gas? The relationship between heat exchange with the surroundings and work is formulated as the first law of thermodynamics.
Basic Knowledge
Internal energy U [J] is the sum of the total kinetic energy + potential energy of the molecules making up a gas. For a monatomic ideal gas, U = (3/2)nRT (n: amount of substance in moles, R: gas constant, T: absolute temperature).
First law of thermodynamics: When a gas receives heat Q and work W done on it by the surroundings, its internal energy is
ΔU = Q + W (W is work done by the surroundings on the gas)
Alternatively, if “work done by the gas on the surroundings” is written as W' (= −W), then Q = ΔU + W'.
Typical processes:
・Isothermal process: ΔT = 0 → ΔU = 0 → Q = W' (heat absorbed = work)
・Constant-volume process: ΔV = 0 → W' = 0 → Q = ΔU
・Constant-pressure process: Pressure is constant → W' = pΔV, Q = ΔU + pΔV
・Adiabatic process: Q = 0 → ΔU = −W' (temperature decreases during expansion)
Internal energy U (a thermodynamic state quantity of a gas; for an ideal gas, it depends only on temperature)
First law of thermodynamics (the thermodynamic form of the law of conservation of energy: ΔU = Q + W)
Isothermal, constant-volume, constant-pressure, and adiabatic processes (four quasi-static processes)
pV diagram (a diagram with pressure p on the vertical axis and volume V on the horizontal axis; area represents work)
Quasi-static process (quasi-static process) (an ideal process in which the system changes while always maintaining equilibrium)
Further Exploration (Background and Significance)
On a pV diagram, the area enclosed by a cycle (closed curve) equals the net work done by the gas on the surroundings in one cycle. This is a powerful tool for visualizing the efficiency of heat engines.
For a monatomic ideal gas, the molar heat capacities at constant pressure and constant volume are Cp = (5/2)R and Cv = (3/2)R, respectively, and their difference is Cp − Cv = R (Mayer’s relation). The reason the specific heat at constant pressure is larger is that energy is needed for expansion work in addition to the energy used to raise the temperature.
During adiabatic expansion, the gas temperature decreases. This is the physical basis of cloud formation (adiabatic cooling as air rises) and the phenomenon of spray cans becoming cold.
- ΔU = Q + W (W is work done by the surroundings on the gas): Apply the sign convention consistently
- For an ideal gas, U depends only on T → ΔU = 0 in an isothermal process
- Constant-volume process: No work → Q = ΔU
- Adiabatic process: Q = 0 → Temperature decreases during expansion
- Area enclosed by a closed curve on a pV diagram = net work
Points to Note (Common Confusions)
① The signs of “work done by the surroundings on the gas W” and “work done by the gas on the surroundings W'” are opposite. Check the problem’s definition first. ② Q ≠ 0 even in an isothermal process because heat is exchanged with the surroundings to maintain the temperature. ③ An adiabatic process is not an isothermal process. In an adiabatic process (Q = 0), the temperature changes. ④ Since internal energy is a state quantity, ΔU is determined only by the initial and final states, regardless of the path.
Practice
- A gas absorbs 500 J of heat and does 300 J of work on the surroundings. Find the change in internal energy.
- The temperature of a monatomic ideal gas rises by 100 K during a constant-volume process. Find the heat absorbed per mole (R = 8.3 J/(mol·K)).
- A gas does 400 J of work on the surroundings during adiabatic expansion. How does its internal energy change?