To calculate the mass of the helium gas, we can use the Ideal Gas Law equation in the form \( PV = nRT \), where:
- \( P \) is the pressure,
- \( V \) is the volume,
- \( n \) is the number of moles,
- \( R \) is the specific gas constant, and
- \( T \) is the temperature in Kelvin.
We need to convert the temperature from Celsius to Kelvin: \[ T(K) = T(°C) + 273.15 = 10 + 273.15 = 283.15 , K \]
We are given:
- Volume \( V = 0.42 , m^3 \)
- Pressure \( P = 750 , kPa = 750,000 , Pa \) (since \( 1 , kPa = 1,000 , Pa \))
- Gas constant \( R = 2.08 , kJ/kg \cdot K = 2080 , J/kg \cdot K \)
Now we can substitute everything into the Ideal Gas Law rearranged to find the mass \( m \): \[ P V = m R T \] So we can solve for mass \( m \): \[ m = \frac{PV}{RT} \]
Now plug in the values: \[ m = \frac{(750,000 , Pa)(0.42 , m^3)}{(2080 , J/kg \cdot K)(283.15 , K)} \]
Calculating the denominator: \[ 2080 , J/kg \cdot K \times 283.15 , K \approx 588,212 , J/kg \]
Now calculate the mass: \[ m = \frac{315,000}{588,212} \] \[ m \approx 0.535 , kg \]
Thus, the mass of the helium gas is approximately 0.535 kg.