Three moles of a monatomic ideal gas are heated at a...

Three moles of a monatomic ideal gas are heated at a constant volume of 1.50 $\mathrm{m}^{3}$ . The amount of heat added is $5.24 \times 10^{3} \mathrm{J}$ . ( a) What is the change in the temperature of the gas? (b) Find the change in its internal energy. (c) Determine the change in pressure.

Key Concepts

Internal Energy Change in an Ideal Gas

For an ideal gas, the change in internal energy is solely a function of temperature change and can be calculated as Delta U = nCv * Delta T. In a constant volume process, this change in internal energy is equal to the heat transferred, as no work is done.

Heat Capacity at Constant Volume

The heat capacity at constant volume (Cv) is the amount of heat required to raise the temperature of a substance by one degree while keeping its volume fixed. For a monatomic ideal gas, Cv is specifically given by (3/2)R, where R is the universal gas constant, reflecting the degrees of freedom available for energy storage.

Pressure Change Calculation

Under the ideal gas law, when the volume is held constant, any increase in temperature will cause a proportional increase in pressure. The relationship Delta P = nR * (Delta T / V) can be derived from the ideal gas law, allowing the determination of pressure change given the number of moles, gas constant, change in temperature, and constant volume.

Ideal Gas Law

The ideal gas law, given by PV = nRT, relates the pressure, volume, temperature, and number of moles of an ideal gas. It provides the foundational relationship to predict how a change in one state variable (e.g., temperature) affects another (e.g., pressure) when some variables are held constant.

Constant Volume Process

A constant volume process (or isochoric process) is one in which the volume of the system does not change. In such processes, no work is done (since work is the product of pressure change and volume change), and any heat added to the system directly changes its internal energy.

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