Problem Statement
Calculate the pressure exerted by a 4 kg mass of nitrogen gas at a temperature of 15°C if the volume is 0.35 m³. The molecular weight of nitrogen is 28.
Given Data
- Mass of Nitrogen, \(m = 4 \, \text{kg}\)
- Temperature, \(t = 15^\circ\text{C}\)
- Volume, \(V = 0.35 \, \text{m}^3\)
- Molecular Weight of Nitrogen (N₂), \(M = 28 \, \text{g/mol}\)
- Universal Gas Constant, \(R_u = 8.314 \, \text{J/mol·K}\)
Solution
1. Convert Units to SI
We convert the temperature to Kelvin and the molecular weight to kg/mol for consistency.
Temperature Conversion:
Molecular Weight Conversion:
2. Calculate the Number of Moles (n)
The number of moles is the mass of the gas divided by its molar mass.
3. Calculate the Pressure (P) using the Ideal Gas Law
The Ideal Gas Law is given by \(PV = nR_uT\). We can rearrange it to solve for pressure.
The pressure exerted by the nitrogen gas is approximately \( P = 978,102 \, \text{N/m}^2 \) or \( 978.1 \, \text{kPa} \).
Explanation of the Ideal Gas Law
The Ideal Gas Law (\(PV = nRT\)) is a fundamental equation that describes the relationship between the pressure (P), volume (V), number of moles (n), and temperature (T) of a hypothetical “ideal” gas. It is a very good approximation for the behavior of many real gases under a wide range of conditions.
- Number of Moles (n): This represents the amount of substance. One mole contains Avogadro’s number (approximately 6.022 x 10²³) of particles.
- Universal Gas Constant (\(R_u\)): This is a physical constant that is the same for all gases when using moles in the equation.
Physical Meaning
The calculated pressure of 978.1 kPa is the force per unit area that the nitrogen gas molecules exert on the walls of the 0.35 m³ container. This pressure is quite high—it is approximately 9.65 times standard atmospheric pressure (which is about 101.3 kPa).
This calculation is essential for safety and design in engineering. Any container built to hold this gas must be strong enough to withstand this internal pressure at the given temperature without rupturing. It also demonstrates how confining a significant mass of gas (4 kg) into a relatively small volume results in a substantial increase in pressure.
