Question

A \(2-m^{3}\) rigid tank initially contains air whose density is \(1.18 \mathrm{kg} / \mathrm{m}^{3} .\) The tank is connected to a high-pressure supply line through a valve. The valve is opened, and air is allowed to enter the tank until the density in the tank rises to \(5.30 \mathrm{kg} / \mathrm{m}^{3}\). Determine the mass of air that has entered the tank.

Short answer

Answer: 8.24 kg

Step-by-step solution

Write down the given information

We have the initial density (\(ρ_i\)) as \(1.18 kg/m^3\), final density (\(ρ_f\)) as \(5.30 kg/m^3\), and the volume of the tank (V) as \(2m^3\).

Calculate the initial mass of the air in the tank

Using the density formula, we will calculate the initial mass of air(m_i) in the tank: $$ \rho_i = \frac{m_i}{V} $$ Rearrange the formula for the initial mass of air: $$ m_i = \rho_i \times V $$ Now, plug in the given values: $$ m_i = 1.18 \,\text{kg/m}^{3} \times 2 \,\text{m}^{3} $$ Calculate the initial mass of air: $$ m_i = 2.36 \,\text{kg} $$

Calculate the final mass of the air in the tank

Using the density formula, we will calculate the final mass of air(m_f) in the tank: $$ \rho_f = \frac{m_f}{V} $$ Rearrange the formula for the final mass of air: $$ m_f = \rho_f \times V $$ Plug in the given values: $$ m_f = 5.30 \,\text{kg/m}^{3} \times 2 \,\text{m}^{3} $$ Calculate the final mass of air: $$ m_f = 10.6 \,\text{kg} $$

Calculate the mass of air that has entered the tank

The mass of air that entered the tank is the difference between the final mass and the initial mass: $$ \Delta m = m_f - m_i $$ Plug in the calculated values: $$ \Delta m = 10.6 \,\text{kg} - 2.36 \,\text{kg} $$ Calculate the mass of air that entered the tank: $$ \Delta m = 8.24 \,\text{kg} $$ The mass of air that has entered the tank is \(8.24 \,\text{kg}\).