Chemistry - Thermodynamics - I Question with Solution | TestHub

Explain the question:

Determine the correct thermodynamic relationships for an ideal gas expanding adiabatically under the condition $\mathrm{T}\propto {\mathrm{V}}^{-1/2}$.

Concept: Adiabatic relations: ${\mathrm{TV}}^{\gamma -1}=$ const. and $P\propto T^{\frac{\gamma }{\gamma -1}}$.

Solution:

For a reversible adiabatic process involving an ideal gas, temperature and volume follow the relationship ${\mathrm{TV}}^{\gamma -1}=$ constant. The problem specifies $\mathrm{T}\propto {\mathrm{V}}^{-1/2}$ , which can be rewritten as TV^1/2 = constant. By equating the exponents, we get:

⟹ γ - 1 = 1.5 (Statement 1 is correct)

Next, using the pressure-volume relationship PV^γ = constant:

$V\propto P^{-1/\gamma }=P^{-\frac{1}{1.5}}=V\propto P^{-\frac{2}{3}}$ (Statement 2 is correct)

Finally, we relate pressure and temperature using P^1-γ T^γ constant ⟹ $P\propto T^{\frac{\gamma }{\gamma -1}}$

$P\propto T^{\frac{1.5}{1.5-1}}=T\frac{{}^{1.5}}{0.5}$=$P\propto T^3$ (Statement 3 is correct)

Mathematically, the given expansion law satisfies three of the provided statements simultaneously.

Final Answer: Options (A, B & C)