Problem Statement
Solve the thermodynamics problem: Solve the thermodynamics problem: A diatomic ideal gas ($\gamma = 7/5$) at $P_1 = 2 \times 10^5$ Pa, $T_1 = 300$ K expands adiabatically until its pressure drops to $P_2 = 10^5$ Pa. Find $T_2$. ($R = 8.314$ J/mol·K) $\gamma = 1.4$ (diatomic) $P_1 = 2 \times 10^5$ Pa, $T_1 = 300$ K $P_2 = 10^5$ Pa Fo
Given Information
- Temperatures, pressures, volumes, and process type as given
- Universal gas constant $R = 8.314\,\text{J mol}^{-1}\text{K}^{-1}$
- $C_p$ and $C_v$ or $\gamma = C_p/C_v$ as applicable
Physical Concepts & Formulas
The First Law of Thermodynamics $\Delta U = Q – W$ is an energy balance: internal energy increases when heat flows in and decreases when the gas does work. For ideal gases, internal energy depends only on temperature: $\Delta U = nC_v \Delta T$. Different processes have different constraints: isothermal ($T = \text{const}$, $W = nRT\ln(V_f/V_i)$), adiabatic ($Q=0$, $PV^\gamma = \text{const}$), isobaric ($P = \text{const}$, $W = P\Delta V$), isochoric ($V = \text{const}$, $W = 0$). Carnot efficiency sets the upper bound for any heat engine: $\eta = 1 – T_C/T_H$.
- $\Delta U = Q – W$ — First Law
- $PV = nRT$ — Ideal Gas Law
- $W_{\text{isothermal}} = nRT\ln(V_f/V_i)$
- $PV^\gamma = \text{const}$ — adiabatic process
- $\eta_{\text{Carnot}} = 1 – T_C/T_H$ — maximum efficiency
Step-by-Step Solution
Step 1 — Identify the process (isothermal, adiabatic, isobaric, isochoric).
Step 2 — Write the appropriate work expression and compute $W$.
Step 3 — Find $\Delta U = nC_v\Delta T$.
Step 4 — Apply First Law: $Q = \Delta U + W$.
Worked Calculation
Substituting all values with units:
Carnot engine: $T_H = 600\,\text{K}$, $T_C = 300\,\text{K}$:
$$\eta = 1 – \frac{300}{600} = 1 – 0.5 = 0.50 = 50\%$$
Answer
$$\boxed{\eta_{\text{Carnot}} = 1 – \dfrac{T_C}{T_H}}$$
Physical Interpretation
A Carnot efficiency of 50% is the theoretical maximum — no real engine can do better between these temperatures. Real steam turbines achieve ~40%; petrol engines ~25–30%. The gap is due to irreversibilities: friction, heat transfer across finite temperature differences, and non-quasi-static processes. This result, derived purely from the Second Law, set a fundamental limit on the Industrial Revolution.
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