The thermal energy of 1.0 mol of a substance is increased by 1.0 J. What is the temperature change if the system is (a) a monatomic gas, (b) a diatomic gas, and (c) a solid?
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1
Identify the type of substance and the corresponding degrees of freedom. For a monatomic gas, there are 3 translational degrees of freedom. For a diatomic gas, there are 5 degrees of freedom (3 translational and 2 rotational). For a solid, typically, all 3 translational and 3 vibrational degrees of freedom are considered, totaling 6 degrees of freedom.
Use the formula for the molar specific heat capacity at constant volume, $C_V = \frac{f}{2} R$, where $f$ is the degrees of freedom and $R$ is the gas constant (approximately 8.314 J/mol\cdot K).
Calculate the specific heat capacity for each case: (a) monatomic gas, (b) diatomic gas, and (c) solid, using their respective degrees of freedom.
Apply the formula for change in temperature, $\Delta T = \frac{\Delta Q}{n C_V}$, where $\Delta Q$ is the change in thermal energy, $n$ is the number of moles, and $C_V$ is the specific heat capacity at constant volume. Here, $\Delta Q = 1.0 \text{ J}$ and $n = 1.0 \text{ mol}$.
Calculate the temperature change $\Delta T$ for each case using the specific heat capacities found in step 3 and the formula from step 4.