A gas has a compressibility factor of 0.5 and a molar volume of 0.4 dm3 mol−1 at a temperature of 800K and pressure x atm. If it shows ideal gas behavior at the same temperature and pressure, the molar volume will be y dm3 mol−1. The value of \(\frac{x}{y}\) is ___. [Use: Gas constant, R = 8 × 10−2 L atm K−1 mol−1]
Correct Answer: 100
Solution and Explanation
Compressibility Factor and Pressure Calculation
The compressibility factor \( Z \) is given by: \[ Z = \frac{V_{\text{real}}}{V_{\text{ideal}}} = 0.5 \]
The real volume \( V_{\text{real}} \) is: \[ V_{\text{real}} = 0.4 \, \text{dm}^3 \, \text{mol}^{-1} = 0.4 \, \text{L/mol} \]
Using the equation for ideal volume: \[ V_{\text{ideal}} = \frac{0.4}{0.5} = 0.8 \, \text{L/mol} \]
Therefore, \( y = 0.8 \, \text{L/mol} \).
Using the ideal gas law, \( PV = nRT \), we calculate the pressure \( P \): \[ P = \frac{1 \times 8 \times 10^{-2} \times 800}{0.8} \]
Solving for \( P \), we get \( P = 80 \, \text{atm} \).
Finally, the ratio \( \frac{x}{y} \) is: \[ \frac{x}{y} = \frac{80}{0.8} = 100 \]
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Concepts Used:
Equilibrium Constant
The equilibrium constant may be defined as the ratio between the product of the molar concentrations of the products to that of the product of the molar concentrations of the reactants with each concentration term raised to a power equal to the stoichiometric coefficient in the balanced chemical reaction.
The equilibrium constant at a given temperature is the ratio of the rate constant of forwarding and backward reactions.
Equilibrium Constant Formula:
Kequ = kf/kb = [C]c [D]d/[A]a [B]b = Kc
where Kc, indicates the equilibrium constant measured in moles per litre.
For reactions involving gases: The equilibrium constant formula, in terms of partial pressure will be:
Kequ = kf/kb = [[pC]c [pD]d]/[[pA]a [pB]b] = Kp
Where Kp indicates the equilibrium constant formula in terms of partial pressures.
- Larger Kc/Kp values indicate higher product formation and higher percentage conversion.
- Lower Kc/Kp values indicate lower product formation and lower percentage conversion.
Medium Kc/Kp values indicate optimum product formation.
Units of Equilibrium Constant:
The equilibrium constant is the ratio of the concentrations raised to the stoichiometric coefficients. Therefore, the unit of the equilibrium constant = [Mole L-1]△n.
where, ∆n = sum of stoichiometric coefficients of products – a sum of stoichiometric coefficients of reactants.