One mole of an ideal mono-atomic gas initially at 300
C is reversibly and adiabatically
expanded to 3 times its initial volume. Determine dq, dw, ΔU, ΔH, ΔS, ΔG and ΔA.
To determine the values of dq (heat transfer), dw (work done), ΔU (change in internal energy), ΔH (change in enthalpy), ΔS (change in entropy), ΔG (change in Gibbs free energy), and ΔA (change in Helmholtz free energy), we need to use the relevant equations and thermodynamic definitions. Let's calculate each value step by step:
1. dq (heat transfer):
Since the expansion is adiabatic, there is no heat transfer involved (dq = 0).
2. dw (work done):
To find the work done, we can use the equation: dw = -PdV. Since the gas expansion is reversible, the process occurs at each step in equilibrium. For an ideal gas, we have the equation PdV = nRdT, where n is the number of moles, R is the gas constant, and dT is the change in temperature.
Here, the initial temperature is given as 300 C, which needs to be converted to Kelvin: T1 = 300 C + 273 = 573 K.
The final temperature can be calculated using the adiabatic expansion process: T2 = T1(V1/V2)^(γ-1), where γ is the heat capacity ratio (Cp/Cv) for a monoatomic ideal gas (~1.67).
Given that the volume expands to 3 times its initial volume (V2 = 3V1), we can calculate T2.
3. ΔU (change in internal energy):
For an adiabatic process, ΔU = dw (since dq = 0).
4. ΔH (change in enthalpy):
For an adiabatic process, ΔH = ΔU (since dq = 0).
5. ΔS (change in entropy):
To calculate the change in entropy, we can use the equation: ΔS = nCv ln(T2/T1). Here, Cv is the molar heat capacity at constant volume, given as (3/2)R for a monoatomic gas.
6. ΔG (change in Gibbs free energy):
For an adiabatic process, ΔG = ΔH (since dq = 0).
7. ΔA (change in Helmholtz free energy):
For an adiabatic process, ΔA = ΔU (since dq = 0).
By following these calculations, we can determine the values of dq = 0, dw, ΔU = dw, ΔH = ΔU, ΔS, ΔG = ΔH, and ΔA = ΔU.