Question

v1 = 84 cu/ft
p1 = 300 psi
t1 = 48°c

v2 = ???? cu/ft
p2 = 110 psi
t2 = 30°c

Explanation:

Step1: Convert temperatures to Kelvin

To use the ideal gas law for combined gas law, temperatures must be in Kelvin. The formula to convert Celsius to Kelvin is \( T(K) = T(^\circ C) + 273.15 \).

For \( T_1 \): \( T_1 = 48 + 273.15 = 321.15 \, K \)

For \( T_2 \): \( T_2 = 30 + 273.15 = 303.15 \, K \)

Step2: Apply the Combined Gas Law

The combined gas law is \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \). We need to solve for \( V_2 \).

Rearranging the formula for \( V_2 \): \( V_2 = \frac{P_1 V_1 T_2}{P_2 T_1} \)

Substitute the known values: \( P_1 = 300 \, psi \), \( V_1 = 84 \, cu/ft \), \( T_1 = 321.15 \, K \), \( P_2 = 110 \, psi \), \( T_2 = 303.15 \, K \)

\( V_2 = \frac{300 \times 84 \times 303.15}{110 \times 321.15} \)

First, calculate the numerator: \( 300 \times 84 = 25200 \); \( 25200 \times 303.15 = 25200 \times 303.15 = 7,649,380 \) (approximate, more accurately \( 300\times84 = 25200; 25200\times303.15 = 25200\times300 + 25200\times3.15 = 7,560,000 + 79,380 = 7,639,380 \))

Denominator: \( 110 \times 321.15 = 35,326.5 \)

Now divide numerator by denominator: \( V_2 = \frac{7,639,380}{35,326.5} \approx 216.25 \, cu/ft \) (Wait, that seems high. Wait, maybe I made a miscalculation. Let's recalculate:

Wait, \( 300\times84 = 25200 \); \( 25200\times303.15 = 25200\times303.15 \). Let's do 25200300 = 7,560,000; 252003.15 = 252003 + 252000.15 = 75,600 + 3,780 = 79,380. So total numerator is 7,560,000 + 79,380 = 7,639,380.

Denominator: 110321.15 = 110321 + 110*0.15 = 35,310 + 16.5 = 35,326.5.

Now divide 7,639,380 by 35,326.5: Let's divide numerator and denominator by 15: 7,639,380 ÷15=509,292; 35,326.5 ÷15=2,355.1. Wait, maybe better to use decimal division: 7,639,380 ÷ 35,326.5. Let's multiply numerator and denominator by 2 to eliminate decimal: 15,278,760 ÷ 70,653 ≈ 216.25. Wait, but let's check the formula again. The combined gas law is \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \), so solving for \( V_2 \) is \( V_2 = V_1 \times \frac{P_1}{P_2} \times \frac{T_2}{T_1} \). So \( \frac{P_1}{P_2} = 300/110 ≈ 2.727 \), \( \frac{T_2}{T_1} = 303.15/321.15 ≈ 0.944 \). Then \( V_2 = 84 \times 2.727 \times 0.944 \). 842.727 ≈ 229.07, 229.070.944 ≈ 216.25. So that's correct.

Answer:

\( V_2 \approx 216.25 \, cu/ft \) (or more precisely, after exact calculation: \( \frac{300 \times 84 \times 303.15}{110 \times 321.15} = \frac{300 \times 84 \times 303.15}{110 \times 321.15} \). Let's simplify fractions: 300/110 = 30/11, 84 is 84, 303.15/321.15 = 30315/32115 = divide numerator and denominator by 15: 2021/2141 (approx). So 30/11 84 2021/2141. 3084=2520, 2520/11 ≈ 229.09, 229.092021/2141 ≈ 229.09*0.944 ≈ 216.2. So the approximate value is 216 cu/ft (or more precisely 216.25).)