Difficulty: Intermediate | Prerequisites: Ch. 11 (energy, first law of thermodynamics)
An ideal gas is a simplified model where molecules bounce around randomly and interact only through elastic collisions. The ideal gas law (PV = NkT or PV = nRT) connects pressure, volume, temperature, and the number of particles. Different thermodynamic processes (isobaric, isochoric, isothermal, adiabatic) hold different variables constant, changing how heat, work, and internal energy relate to each other.
Ideal gas
A theoretical gas composed of many randomly moving point particles that interact only through perfectly elastic collisions. Real gases behave approximately as ideal gases at low pressure and high temperature.
Boltzmann's constant (k_B)
k_B = 1.38 × 10⁻²³ J/K. Connects temperature to the energy of individual molecules. Think of it as the conversion factor between "temperature in kelvin" and "energy per particle."
Ideal gas law
PV = NkT (microscopic form, N = number of particles) or PV = nRT (macroscopic form, n = number of moles, R = 8.31 J/(mol·K)). Relates pressure, volume, temperature, and amount of gas.
Thermal energy of an ideal gas
E_th = (3/2)Nk_BT = (3/2)nRT. The thermal energy depends only on temperature for an ideal gas. In simple terms, hotter gas has more thermal energy, proportional to the absolute temperature.
Root-mean-square speed (v_rms)
v_rms = √(3k_BT/m), where m is the mass of one molecule. This is the speed of a "typical" atom in the gas. It is not the average speed, but it is close.
Pressure
Force per unit area: P = F/A. In a gas, pressure arises from molecules bouncing off the container walls.
Gauge pressure
The pressure above atmospheric pressure: P_gauge = P - 1 atm. What most pressure gauges actually read.
Isobaric process
A process at constant pressure. P_f = P_i.
Isochoric process
A process at constant volume. V_f = V_i. Also called isovolumetric.
Isothermal process
A process at constant temperature. On a PV diagram it appears as a hyperbola, because P = nRT/V and T is fixed.
Adiabatic process
A process with no heat transfer (Q = 0). Can involve expansion (cooling) or compression (heating). Faster than isothermal on a PV diagram.
Molar specific heat at constant volume (C_v)
C_v = (3/2)R for a monatomic ideal gas. The heat needed per mole per kelvin when volume is held fixed.
Molar specific heat at constant pressure (C_p)
C_p = (5/2)R for a monatomic ideal gas. The heat needed per mole per kelvin when pressure is held fixed. Always larger than C_v because some energy goes to doing work (expanding against constant pressure).
Microscopic picture
A gas is made of a large number of particles (atoms or molecules) in constant random motion
Temperature is a measure of the average kinetic energy per particle: E_avg = (3/2)k_BT
Thermal energy of the whole gas: E_th = N × (3/2)k_BT = (3/2)NkT
For two containers at different temperatures: if container A has a temperature change ΔT, its thermal energy changes by ΔE_th proportional to ΔT. A container with twice the temperature change has twice the thermal energy change.
Pressure from the molecular viewpoint
Pressure arises from molecules striking the walls of the container
P = F/A (force per unit area)
The ideal gas law, PV = NkT, follows from averaging the momentum transferred by molecular collisions
Unit conversions for pressure
1 atm = 1.013 × 10⁵ Pa = 101.3 kPa
1 Pa = 1 N/m²
Always convert to kelvin for gas law calculations: T(K) = T(°C) + 273
Speed of molecules
v_rms = √(3k_BT / m), where m = mass of one molecule
Heavier molecules move more slowly at the same temperature
The number of moles n relates to the number of particles N: N = nN_A, where N_A = 6.02 × 10²³ (Avogadro's number)
You should be able to read PV diagrams fluently. A process from one state to another on a PV diagram is called a trajectory. Each type of process has a distinctive shape.
Isobaric process (constant pressure)
Horizontal line on a PV diagram
P_f = P_i
Work done by the gas: W_by gas = P(ΔV)
There is both work and heat exchange: Q = ΔE_th + W_on, or equivalently Q = ΔE_th - W_by
Isochoric process (constant volume)
Vertical line on a PV diagram
V_f = V_i
No work done (W = 0), because the volume does not change. For work to happen, volume must change.
All heat goes to changing thermal energy: Q = ΔE_th
P ∝ T at constant volume
Isothermal process (constant temperature)
A hyperbola on the PV diagram, because P = nRT/V with T constant, so PV = constant
ΔT = 0, therefore ΔE_th = 0 (thermal energy depends only on temperature for an ideal gas)
Q = -W_on = W_by gas. All heat input goes to work output, and vice versa.
Adiabatic process (no heat transfer)
Q = 0
ΔE_th = W (thermal energy changes only through work)
Adiabatic expansion: the gas does work, loses thermal energy, cools down
Adiabatic compression: work is done on the gas, it gains thermal energy, heats up
The curve on a PV diagram is steeper than an isothermal curve
TV^(γ-1) = constant, where γ = C_p/C_v
Solving for Q in any process
For any process: Q = ΔE_th - W_on. If the gas is expanding (ΔV > 0), the gas does positive work on its surroundings, so W_on is negative. Track signs carefully.
Remember: for an isochoric process, Q = ΔE_th = (3/2)nRΔT.
Molar specific heat at constant volume
C_v = (3/2)R for a monatomic ideal gas
Q = nC_vΔT (when volume is constant)
Q = ΔE_th (since no work is done)
Molar specific heat at constant pressure
At constant pressure, there is both work and heat exchange
ΔE_th = Q - W_by, and W_by = PΔV = nRΔT
So Q = ΔE_th + PΔV = (3/2)nRΔT + nRΔT = (5/2)nRΔT
Therefore C_p = (5/2)R
Q = nC_pΔT (when pressure is constant)
Why C_p is always larger than C_v
At constant pressure, the gas expands as it heats, doing work on its surroundings. You need to supply extra heat to cover that work on top of raising the temperature. At constant volume, all the heat goes directly to raising the temperature.
The relationship: C_p = C_v + R
Worked example pattern
A closed system undergoing two separate processes (A and B)
For each process, identify whether it is isobaric or isochoric
Apply the correct Q formula: Q = nC_vΔT (isochoric) or Q = nC_pΔT (isobaric)
The source notes flag this as a common exam pattern: if you have trouble knowing where to start, look at the derivations of C_v and C_p above
PV = Nk_BT = nRTE_{th} = \frac{3}{2}Nk_BT = \frac{3}{2}nRTv_{rms} = \sqrt{\frac{3k_BT}{m}}W_{by\ gas} = P \Delta V \quad \text{(isobaric)}Q = nC_v \Delta T \quad \text{(isochoric, } C_v = \tfrac{3}{2}R\text{)}Q = nC_p \Delta T \quad \text{(isobaric, } C_p = \tfrac{5}{2}R\text{)}C_p = C_v + R1\ \text{atm} = 1.013 \times 10^5\ \text{Pa}R = 8.31\ \text{J/(mol·K)} \quad ; \quad k_B = 1.38 \times 10^{-23}\ \text{J/K}Car engines run through a cycle of isobaric, isochoric, and adiabatic processes. Understanding PV diagrams lets you calculate the theoretical efficiency of any engine cycle.
Scuba divers use the ideal gas law constantly: as a diver descends, pressure increases, and the gas in their tank and lungs compresses accordingly. Boyle's law (isothermal, PV = constant) governs this.
Students often think temperature measures the total energy of a gas. It does not. Temperature measures the average energy per particle. A large container of cool gas can have more total thermal energy than a small container of hot gas.
Students frequently forget to convert to kelvin before using the ideal gas law. Celsius and Fahrenheit do not work in PV = nRT.
Students sometimes assume that in an isothermal process, nothing changes. In fact, pressure and volume both change; only temperature (and therefore thermal energy) stays constant.
Students mix up C_v and C_p, or forget why C_p > C_v. The key: at constant pressure the gas does work as it expands, so you need more heat input to achieve the same temperature rise.
⚠️ PV diagram interpretation is heavily tested. Know the shape of each process (horizontal = isobaric, vertical = isochoric, hyperbola = isothermal, steep curve = adiabatic).
⚠️ The example problem with a closed system undergoing two separate processes (A and B) is flagged in the source notes as very useful. Know how to identify each process type and apply the right Q formula.
⚠️ Always convert temperature to kelvin. This is a free-mark issue: get the conversion wrong and everything downstream is wrong.
⚠️ Know the derivation of C_p from C_v. The exam may ask you to show why C_p = C_v + R.
True or False: The thermal energy of an ideal gas depends on both temperature and volume. (False. It depends only on temperature: E_th = (3/2)nRT.)
Fill in the blank: In an isochoric process, W = ______ because ______ does not change. (0, volume)
True or False: An isothermal expansion of an ideal gas increases its thermal energy. (False. ΔT = 0, so ΔE_th = 0.)
Fill in the blank: C_p is always ______ than C_v by an amount equal to ______. (larger, R)
True or False: Gauge pressure includes atmospheric pressure. (False. Gauge pressure = total pressure minus atmospheric pressure.)
Q: A monatomic ideal gas undergoes an isochoric process where the temperature rises by 50 K. If there are 2 moles of gas, how much heat is transferred?
A: Q = nC_vΔT = 2 × (3/2)(8.31) × 50 = 2 × 12.465 × 50 = 1246.5 J ≈ 1247 J.
Q: The same gas then undergoes an isobaric expansion with the same ΔT = 50 K. How much heat is transferred now?
A: Q = nC_pΔT = 2 × (5/2)(8.31) × 50 = 2 × 20.775 × 50 = 2077.5 J ≈ 2078 J. More heat is needed because part of it goes to doing work.
Q: An ideal gas is compressed isothermally. Does its thermal energy change? Does its temperature change?
A: Neither changes. Isothermal means constant temperature, and for an ideal gas, thermal energy depends only on temperature. The work done on the gas is exactly balanced by heat flowing out.
Q: Why is the adiabatic curve on a PV diagram steeper than the isothermal curve?
A: In an adiabatic process, Q = 0, so the gas must change temperature as it expands or compresses. As it expands, it cools (pressure drops faster). In an isothermal process, heat flows in to maintain temperature, so the pressure drops more gently.
Q: Convert 200 kPa to atm.
A: 200 kPa / 101.3 kPa per atm ≈ 1.97 atm.
The ideal gas law builds on the temperature and thermal energy concepts from Ch. 11.3. Temperature is the average kinetic energy per particle; the gas law makes that quantitative.
Gas processes (isobaric, isochoric, isothermal, adiabatic) are direct applications of the first law of thermodynamics from Ch. 11.4. Each process constrains a different variable, giving a different relationship between Q, W, and ΔE_th.
Specific heat of gases connects to the specific heat and calorimetry material in Ch. 12.5 and 12.6, extending those ideas from solids and liquids to gases.
Ideal gas law, PV = nRT, Boyle's law, Charles's law, Gay-Lussac's law, isobaric, isochoric, isovolumetric, isothermal, adiabatic, PV diagram, Boltzmann constant, root-mean-square speed, v_rms, molar specific heat, Cv, Cp, monatomic gas, kinetic theory, gas pressure, gauge pressure, kelvin, PHYS 115, Physics 115 Exam 1, thermodynamics, ideal gas processes