Question

which term expresses the pressure of a mixture of two ideal gases, a and b?
(na + nb)kt/v
xa pa + xb pb
pv/nt
ptot/(xa + xb)

Explanation:

Step1: Recall Ideal Gas Law and Dalton's Law

The ideal gas law is \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is moles, \( R \) is gas constant, \( T \) is temperature. For a mixture of ideal gases, Dalton's law of partial pressures states that total pressure \( P_{total} \) is the sum of partial pressures (\( P_A + P_B \)). Also, the total moles \( n_{total}=n_A + n_B \).

Step2: Analyze Each Option

• Option 1: \((n_A + n_B)\frac{RT}{V}\)

From ideal gas law \( P = \frac{nRT}{V} \). For total moles \( n = n_A + n_B \), total pressure \( P_{total}=\frac{(n_A + n_B)RT}{V} \). This matches the total pressure of the mixture (since total moles are sum of moles of A and B, and using ideal gas law for the mixture).

• Option 2: \( X_A P_A + X_B P_B \)

Partial pressure is \( P_i = X_i P_{total} \), so \( X_A P_A + X_B P_B = X_A(X_A P_{total}) + X_B(X_B P_{total}) \), which is not correct. The correct form for total pressure from partial pressures is \( P_{total}=P_A + P_B \), or \( P_{total}=X_A P_{A}^{\circ}+X_B P_{B}^{\circ} \) (for vapor pressure, not here). So this is incorrect.

• Option 3: \( \frac{PV}{nT} \)

Rearranging ideal gas law \( PV = nRT \) gives \( R=\frac{PV}{nT} \), which is the gas constant, not total pressure of the mixture. Incorrect.

• Option 4: \( \frac{P_{total}}{X_A + X_B} \)

Mole fractions \( X_A + X_B = 1 \) (since \( X_i=\frac{n_i}{n_{total}} \), sum of mole fractions is 1). So \( \frac{P_{total}}{1}=P_{total} \), which is trivial and not an expression for total pressure of the mixture (it just gives \( P_{total} \) back, not a way to calculate it from A and B). Incorrect.

Answer:

\((n_A + n_B)\frac{RT}{V}\) (the first option)