One mole of a monoatomic ideal gas undergoes a thermodynamic process such that its pressure is directly proportional to its volume. If the gas is heated so that its temperature increases from T₀ to 4T₀ , the total heat supplied to the gas is:

Step-by-step solution

For the given process, P V , which means P = kV . Using the ideal gas equation PV = nRT , the work done can be written as: W = P dV = _ V₁ ^ V₂ kV dV = 1 2 k(V₂^2 - V₁^2) Since PV = kV^2 = nRT , we have k(V₂^2 - V₁^2) = nR(T₂ - T₁) . Thus, the work done is W = 1 2 nR(T₂ - T₁) . Given n = 1 mole, T₁ = T₀ , and T₂ = 4T₀ : W = 1 2 (1)R(4T₀ - T₀) = 1.5 R T₀ The change in internal energy for a monoatomic gas is: U = nC_v T = (1) ( 3 2 R )(4T₀ - T₀) = 4.5 R T₀ According to the first law of thermodynamics, the total heat