Question

how many moles of gas must be forced into a 5.2 l tire to give it a gauge pressure of 30.4 psi at 22 °c? the gauge pressure is relative to atmospheric pressure. assume that atmospheric pressure is 14.6 psi so that the total pressure in the tire is 45.0 psi. express your answer using two significant figures.

Explanation:

Step1: Determine total pressure

Gauge pressure is 30.4 psi, atmospheric pressure is 14.6 psi. Total pressure \( P = 30.4 + 14.6 = 45.0 \) psi. Convert to atm: \( 1 \) atm \( = 14.696 \) psi, so \( P = \frac{45.0}{14.696} \approx 3.062 \) atm.

Step2: Convert temperature to Kelvin

\( T = 22 + 273.15 = 295.15 \) K, volume \( V = 5.2 \) L, \( R = 0.0821 \) L·atm/(mol·K).

Step3: Apply ideal gas law \( PV = nRT \)

Solve for \( n \): \( n = \frac{PV}{RT} \). Substitute values: \( n = \frac{3.062 \times 5.2}{0.0821 \times 295.15} \). Calculate numerator: \( 3.062 \times 5.2 \approx 15.9224 \). Denominator: \( 0.0821 \times 295.15 \approx 24.232 \). Then \( n \approx \frac{15.9224}{24.232} \approx 0.657 \) mol. Round to two significant figures: \( n \approx 0.66 \) mol.

Answer:

\( 0.66 \)