Problem 2.28 — Isochoric Heating: Heat and Internal Energy

Problem Statement

Solve the work-energy problem: Two moles of diatomic gas ($\gamma=1.4$) are heated at constant volume from 200 K to 500 K. Find $Q$ and $\Delta U$. For a diatomic gas: $C_v = \frac{5}{2}R$. At constant volume $W=0$, so $Q=\Delta U$. $$Q = \Delta U = \nu C_v\Delta T = 2.0\times\frac{5}{2}\times8.314\times300 = 12{,}471\ \text{J} \

Given Information

• See problem statement for all given quantities.

Physical Concepts & Formulas

Thermodynamics governs energy transformations involving heat and work. The First Law $\Delta U = Q – W$ expresses energy conservation. For an ideal gas, internal energy depends only on temperature ($U = nC_VT$), and the equation of state $PV = nRT$ links pressure, volume, and temperature.

• $\Delta U = Q – W$ — First Law of Thermodynamics
• $PV = nRT$ — ideal gas equation
• $C_P – C_V = R$, $\gamma = C_P/C_V$
• $W = \int P\,dV$ — work done by gas

Step-by-Step Solution

Step 1 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.

Step 2 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.

Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.

Worked Calculation

$$Q = \Delta U = \nu C_v\Delta T = 2.0\times\frac{5}{2}\times8.314\times300 = 12{,}471\ \text{J} \

Given Information

• Mass $m$, velocity $v$, height $h$, or other given quantities
• Any forces doing work (conservative or non-conservative) as specified

Physical Concepts & Formulas

The Work-Energy Theorem states that the net work done on an object equals its change in kinetic energy: $W_{\text{net}} = \Delta KE$. For conservative forces (gravity, spring, electric), a potential energy function $U$ exists such that $W = -\Delta U$, and the total mechanical energy $E = KE + U$ is conserved. Non-conservative forces (friction, air drag) remove mechanical energy, converting it to thermal energy. The power delivered is $P = dW/dt = \vec{F}\cdot\vec{v}$.

• $W = \vec{F}\cdot\vec{d} = Fd\cos\theta$ — work done by constant force
• $KE = \frac{1}{2}mv^2$ — kinetic energy
• $U_g = mgh$ — gravitational PE (near Earth’s surface)
• $U_s = \frac{1}{2}kx^2$ — elastic PE
• $W_{\text{net}} = \Delta KE = KE_f – KE_i$ — work-energy theorem
• $E_i = E_f$ (when only conservative forces act)

Step-by-Step Solution

Step 1 — Identify all forces and whether they are conservative.

Step 2 — Apply conservation of energy (if no friction):

$$

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Step 3 — If friction acts:

$$

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Step 4 — Solve for the unknown (usually $v_f$ or $d$).

Worked Calculation

Substituting all values with units:

Ball of mass $m = 0.5\,\text{kg}$ dropped from $h = 10\,\text{m}$:

$$

Answer

$$\boxed{v_f = \sqrt{2g h}}$$

Physical Interpretation

The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.