Problem Statement
Define the Gruneisen parameter $\Gamma = V(\partial p/\partial U)_V$ and show it relates thermal pressure to internal energy in solids.
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Identify given quantities and set up the problem: The Gruneisen parameter:
Step 2 — Apply the relevant physical law or equation: $$\Gamma = V\left(\frac{\partial p}{\partial U}\right)_V = \frac{V\alpha_V}{\kappa_T C_v}$$
Step 3 — Solve algebraically for the unknown: where $\alpha_V = \frac{1}{V}(\partial V/\partial T)_p$ is the volumetric thermal expansion coefficient and $\kappa_T = -\frac{1}{V}(\partial V/\partial p)_T$ is the isothermal compressibility.
Step 4 — Substitute numerical values with units: Physical meaning: $\Gamma$ relates the increase in vibrational frequency of lattice modes to volume compression. It determines how strongly thermal vibrations contribute to pressure:
Step 5 — Compute and check the result: $$\Delta p_{thermal} = \Gamma\frac{\Delta U}{V}$$
Step 6: For most metals, $\Gamma \approx 1.5$–$2.5$. For an ideal monatomic gas, $\Gamma = 2/3$ (since $pV = \frac{2}{3}U$).
Worked Calculation
$$\Gamma = V\left(\frac{\partial p}{\partial U}\right)_V = \frac{V\alpha_V}{\kappa_T C_v}$$
$$\Delta p_{thermal} = \Gamma\frac{\Delta U}{V}$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
The Gruneisen parameter:
$$\Gamma = V\left(\frac{\partial p}{\partial U}\right)_V = \frac{V\alpha_V}{\kappa_T C_v}$$
where $\alpha_V = \frac{1}{V}(\partial V/\partial T)_p$ is the volumetric thermal expansion coefficient and $\kappa_T = -\frac{1}{V}(\partial V/\partial p)_T$ is the isothermal compressibility.
Physical meaning: $\Gamma$ relates the increase in vibrational frequency of lattice modes to volume compression. It determines how strongly thermal vibrations contribute to pressure:
$$\Delta p_{thermal} = \Gamma\frac{\Delta U}{V}$$
For most metals, $\Gamma \approx 1.5$–$2.5$. For an ideal monatomic gas, $\Gamma = 2/3$ (since $pV = \frac{2}{3}U$).
Answer
$$\boxed{\Delta p_{thermal} = \Gamma\frac{\Delta U}{V}}$$
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.
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