An ideal gas undergoes a process in which its pressure P is directly proportional to the cube of its density . The volume expansion coefficient of the gas as a function of temperature T is :

Step-by-step solution

Given that P ^3 . Since the density of a given mass of gas is inversely proportional to its volume ( 1 V ), we have: P 1 V^3 P V^3 = constant = K From the ideal gas equation, P = nRT V . Substituting this into the process equation: ( nRT V ) V^3 = K T V^2 = K nR = C (a new constant) Rearranging for V as a function of T : V^2 = C T⁻¹ V = C T^ -1/2 Differentiating with respect to T : dV dT = C (- 1 2 ) T^ -3/2 The volume expansion coefficient is defined as = 1 V dV dT . Substituting V and dV dT : = C (- 1 2 ) T^ -3/2