Question

Nitrogen gas is contained in a rigid 1-m tank, initially at 10 bar, \(300 \mathrm{~K}\). Heat transfer to the contents of the tank occurs until the temperature has increased to \(400 \mathrm{~K}\). During the process, a pressure-relief valve allows nitrogen to escape, maintaining constant pressure in the tank. Neglecting kinetic and potential energy effects, and using the ideal gas model with constant specific heats evaluated at \(350 \mathrm{~K}\), determine the mass of nitrogen that escapes, in \(\mathrm{kg}\), and the amount of energy transfer by heat, in \(\mathrm{kJ}\).

Short answer

The mass of nitrogen that escapes is 0.2808 kg, and the energy transfer by heat is 62.57 kJ.

Step-by-step solution

Understand the initial conditions

The nitrogen is initially at 10 bar and 300 K in a 1-m tank. The initial mass of nitrogen can be determined using the ideal gas law: \[ PV = nRT \]

Determine initial mass using ideal gas law

Rewriting the ideal gas law to solve for the number of moles, we get: \[ n_1 = \frac{{P_1 V}}{{R T_1}} \]Where: \( P_1 = 10 \text{ bar} = 10 \times 10^5 \text{ Pa} \), \( V = 1 \text{ m}^3 \), \( R = 8.314 \text{ J/(mol·K)} \), \( T_1 = 300 \text{ K} \)

Calculate the initial number of moles

Substitute the values into the equation: \[ n_1 = \frac{{10 \times 10^5 \times 1}}{{8.314 \times 300}} = 40.13 \text{ mol} \]

Relate moles to mass

Convert the number of moles to mass using the molar mass of nitrogen \( \text{M} = 28 \text{ g/mol} \): \[ m_1 = n_1 \times \text{M} = 40.13 \times 28 \times 10^{-3} = 1.12364 \text{ kg} \]

Understand final conditions and calculate final moles

The final temperature is 400 K, with pressure maintained at 10 bar. Use the ideal gas law again for the final conditions: \[ n_2 = \frac{{P V}}{{R T_2}} = \frac{{10 \times 10^5 \times 1}}{{8.314 \times 400}} = 30.10 \text{ mol} \]

Calculate the final mass

Convert the final number of moles to mass: \[ m_2 = n_2 \times \text{M} = 30.10 \times 28 \times 10^{-3} = 0.8428 \text{ kg} \]

Determine the mass of nitrogen that escapes

Find the difference in mass from initial to final state: \[ m_{escaped} = m_1 - m_2 = 1.12364 - 0.8428 = 0.2808 \text{ kg} \]

Calculate the energy transfer by heat using the first law of thermodynamics

The energy transfer by heat can be computed from the change in internal energy: \[ Q = \Delta U = m \cdot c_v \cdot \Delta T \]Using specific heat at constant volume for nitrogen \( c_v \approx 0.743 \text{ kJ/(kg·K)} \), and the mass of nitrogen remaining in the tank: \( m = 0.8428 \text{ kg} \) and temperature change \( \Delta T = 400 - 300 = 100 \text{ K} \). Thus, \[ Q = 0.8428 \times 0.743 \times 100 = 62.57 \text{ kJ} \]

Key concepts

Ideal gas law

The ideal gas law connects the physical properties of gases such as pressure, volume, temperature, and number of moles of gas. The general formula for the ideal gas law is \[ PV = nRT \], where:

• \(P\): Pressure
• \(V\): Volume
• \(n\): Number of moles
• \( R\): Universal gas constant
• \(T\): Temperature

In this exercise, you used the ideal gas law to calculate both the initial and final masses of nitrogen gas. By knowing the initial conditions (pressure, volume, temperature) and applying the formula, the number of moles \( n_1\) and \( n_2 \) can be determined. These moles relate directly to mass when multiplied by the molar mass of nitrogen (28 g/mol). This step is crucial as it sets up the calculation for the amount of nitrogen that escapes due to the change in temperature.

Heat transfer

Heat transfer refers to the movement of thermal energy from one place to another. In this scenario, heat is transferred to the nitrogen gas, causing its temperature to rise from 300 K to 400 K. Heat transfer can be calculated using the formula \[ Q = mc_v\triangle T \], where:

• \(Q\): Energy transfer by heat (in kJ)
• \( m \): Mass remaining in the tank
• \( c_v \): Specific heat at constant volume (for nitrogen given as 0.743 kJ/(kg·K))
• \( \triangle T \): Change in temperature

By understanding these elements, you can determine how much energy is required to increase the temperature of the nitrogen gas, keeping the pressure constant via the relief valve mechanism.

First law of thermodynamics

The first law of thermodynamics, also known as the law of energy conservation, states that energy cannot be created or destroyed, only transformed or transferred. The formula to express this is: \[ \triangle U = Q - W \], where

• \( \triangle U \): Change in internal energy
• \( Q \): Heat added to the system
• \( W \): Work done by the system

In this exercise, you simplified the problem by neglecting the kinetic and potential energy effects, allowing you to use \[ Q = \triangle U \]. This was useful for calculating the energy transfer by heat straightforwardly by knowing the change in internal energy which depends on mass, specific heat, and temperature change. This reflects how energy input via heat translates directly to a change in internal energy, as observed with the nitrogen gas heating.