Ranking: heat in isobaric and isochoric processes

Problem

Consider three processes on $1$ mole of a monatomic ideal gas.

• A: isobaric heating from $300\;\text{K}$ to $400\;\text{K}$.
• B: isochoric heating from $300\;\text{K}$ to $400\;\text{K}$.
• C: isobaric heating from $300\;\text{K}$ to $500\;\text{K}$.

Rank the heat exchanged from smallest to largest.

Solution

Setup. Monatomic gas: $f = 3$, so $C_v = \tfrac{3}{2}R$ and $C_p = \tfrac{5}{2}R$. With $n = 1\;\text{mol}$ and $R = 8{,}31\;\text{J/(mol·K)}$:

• isochoric: $Q = n C_v \Delta T$;
• isobaric: $Q = n C_p \Delta T$.

Process B (isochoric, $\Delta T = 100\;\text{K}$). $Q_B = n C_v \Delta T = \tfrac{3}{2}\cdot 8{,}31 \cdot 100 = 1247\;\text{J}$

Process A (isobaric, $\Delta T = 100\;\text{K}$). $Q_A = n C_p \Delta T = \tfrac{5}{2}\cdot 8{,}31 \cdot 100 = 2078\;\text{J}$

Process C (isobaric, $\Delta T = 200\;\text{K}$). $Q_C = n C_p \Delta T = \tfrac{5}{2}\cdot 8{,}31 \cdot 200 = 4155\;\text{J}$

Ranking. For the same $\Delta T$, the isobaric process requires more heat than the isochoric one (part of the heat becomes expansion work); C has twice the $\Delta T$ of A: $\ev{Q_B < Q_A < Q_C}$

Links

Topics: Thermodynamics Concepts: Specific heat and heat capacity Skills: Ranking reasoning Objects: Ideal gas