An ideal-gas process is described by p=cV^1/2, where c is a constant. Find an expression for the work done on the gas in this process as the volume changes from V₁ to V₂.

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Start by recalling the formula for work done on a gas during a process: \( W = \int_{V_1}^{V_2} p \, dV \). Here, \( p \) is the pressure, and \( V \) is the volume.

Substitute the given relationship \( p = cV^{1/2} \) into the work formula. This gives \( W = \int_{V_1}^{V_2} cV^{1/2} \, dV \).

Simplify the integral by factoring out the constant \( c \): \( W = c \int_{V_1}^{V_2} V^{1/2} \, dV \).

Evaluate the integral \( \int V^{1/2} \, dV \) using the power rule for integration: \( \int V^n \, dV = \frac{V^{n+1}}{n+1} \), where \( n = 1/2 \). This results in \( \frac{2}{3} V^{3/2} \).

Substitute the limits of integration \( V_1 \) and \( V_2 \) into the evaluated integral: \( W = c \left[ \frac{2}{3} V^{3/2} \right]_{V_1}^{V_2} = \frac{2c}{3} \left( V_2^{3/2} - V_1^{3/2} \right) \). This is the expression for the work done on the gas.

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Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, and temperature of an ideal gas through the equation PV = nRT. In this context, understanding how pressure (p) and volume (V) interact is crucial for analyzing the work done during a gas process. The law assumes that gas particles do not interact and occupy no volume, simplifying calculations in thermodynamic processes.

Work Done by a Gas

In thermodynamics, the work done on or by a gas during a volume change is calculated using the integral of pressure with respect to volume. For a process where pressure is a function of volume, such as p = cV^2, the work done can be expressed as W = ∫(p dV) from V₁ to V₂. This integral captures the area under the pressure-volume curve, representing the energy transferred during the expansion or compression of the gas.

Integration in Physics

Integration is a fundamental mathematical tool used in physics to calculate quantities that accumulate over a continuous range, such as work done over a changing volume. In the context of the given problem, integrating the pressure function with respect to volume allows us to derive the total work done on the gas as it transitions from one volume to another. Mastery of integration techniques is essential for solving many problems in thermodynamics and mechanics.