n moles of an ideal gas at temperature T₁ and volume V₁ expand isothermally until the volume has doubled. In terms of n, T₁, and V₁, what are the work done on the gas?

검증된 단계별 안내

Step 1: Recognize that the process is isothermal, meaning the temperature remains constant throughout the expansion. For an ideal gas undergoing isothermal expansion, the work done on the gas can be calculated using the formula: \( W = -nRT \ln \left( \frac{V_2}{V_1} \right) \), where \( V_2 \) is the final volume, \( V_1 \) is the initial volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature.

Step 2: Substitute \( V_2 = 2V_1 \) into the formula, as the problem states that the volume doubles during the expansion. This gives \( W = -nRT \ln \left( \frac{2V_1}{V_1} \right) \).

Step 3: Simplify the expression inside the logarithm. Since \( \frac{2V_1}{V_1} = 2 \), the formula becomes \( W = -nRT \ln(2) \).

Step 4: Note that the negative sign indicates that work is done by the gas on its surroundings during the expansion, as the gas is expanding against external pressure.

Step 5: Express the final formula for the work done on the gas in terms of the given variables: \( W = -nRT_1 \ln(2) \), where \( T_1 \) is the constant temperature during the isothermal process.

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주요 개념

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Isothermal Process

An isothermal process occurs when a system's temperature remains constant while it undergoes a change in volume. For an ideal gas, this means that any heat added to the system is used to do work, rather than increasing the internal energy. The relationship between pressure, volume, and temperature is described by the ideal gas law, which is crucial for understanding how the gas behaves during expansion.

Work Done by a Gas

The work done by a gas during expansion or compression can be calculated using the formula W = ∫ P dV, where P is the pressure and dV is the change in volume. In an isothermal expansion of an ideal gas, the work done can be expressed as W = nRT ln(Vf/Vi), where Vf and Vi are the final and initial volumes, respectively. This highlights the relationship between work, temperature, and volume changes in the gas.

Ideal Gas Law

The ideal gas law, represented as PV = nRT, relates the pressure (P), volume (V), temperature (T), and number of moles (n) of an ideal gas. This law is fundamental in thermodynamics and allows for the calculation of one variable when the others are known. In the context of isothermal expansion, it helps determine how pressure changes as volume increases while maintaining a constant temperature.