Question

A 38.0-L gas tank at \(35^{\circ} \mathrm{C}\) has nitrogen at a pressure of \(4.65 \mathrm{~atm}\). The contents of the tank are transferred without loss to an evacuated 55.0-L tank in a cold room where the temperature is \(4^{\circ} \mathrm{C}\). What is the pressure in the tank?

Short answer

Answer: The pressure of nitrogen in the gas tank after being transferred to the new environment is approximately 3.07 atm.

Step-by-step solution

Write down the given information

We are given the following information: - Initial volume: \(V_1 = 38.0\,\text{L}\) - Initial temperature: \(T_1 = 35^\circ\text{C}\) - Initial pressure: \(P_1 = 4.65\,\text{atm}\) - Final volume: \(V_2 = 55.0\,\text{L}\) - Final temperature: \(T_2 = 4^\circ\text{C}\)

Convert temperatures to Kelvin

To work with the ideal gas law, we need to express temperatures in Kelvin. To do this, we need to add 273.15 to the Celsius values: \(T_1 = 35^\circ\text{C} + 273.15\,\text{K} = 308.15\,\text{K}\) \(T_2 = 4^\circ\text{C} + 273.15\,\text{K} = 277.15\,\text{K}\)

Find the number of moles of nitrogen

Using the ideal gas law, which is given by \(PV = nRT\), we can find the number of moles of nitrogen in the gas tank. Here, \(n\) is the number of moles, \(R\) is the ideal gas constant (\(0.0821\,\text{L}\,\text{atm/mol}\,\text{K}\)), and \(T\) is the temperature in Kelvin. Rearranging the ideal gas law to solve for \(n\), we get: $$n = \frac{P_1V_1}{RT_1}$$ Substituting the given values: $$n = \frac{4.65\,\text{atm} \cdot 38.0\,\text{L}}{0.0821\,\text{L}\,\text{atm/mol}\,\text{K} \cdot 308.15\,\text{K}}$$ $$n \approx 7.30\,\text{moles}$$

Calculate the final pressure

Now we will substitute the final conditions into the ideal gas law to determine the final pressure \(P_2\): $$P_2V_2 = nRT_2$$ Rearranging the equation to solve for \(P_2\) and substituting the values: $$P_2 = \frac{nRT_2}{V_2}$$ $$P_2 = \frac{7.30\,\text{moles}\cdot 0.0821\,\text{L}\,\text{atm/mol}\,\text{K}\cdot 277.15\,\text{K}}{55.0\,\text{L}}$$ $$P_2 \approx 3.07\,\text{atm}$$

Final Answer

The pressure in the gas tank after being transferred to the new environment is approximately \(3.07\,\text{atm}\).

Key concepts

Gas Pressure Calculation

Understanding gas pressure calculation is vital when dealing with the properties of gases. Pressure is defined as the force applied per unit area, and in the context of gases, it is the result of gas molecules colliding with the walls of their container. The Ideal Gas Law, represented as
\( PV = nRT \),
links pressure (P) with volume (V), the number of moles of gas (n), the temperature (T) in Kelvin, and the ideal gas constant (R).

Application in Calculations

In practical scenarios, such as determining the pressure in a gas tank, you take the initial state of a gas, use the ideal gas law to find out the number of moles, and then apply the final conditions to solve for the new pressure after a change in conditions.
For this calculation, it’s crucial to maintain consistent units throughout, such as litres for volume and atmospheres for pressure. The gas constant R value is chosen based on these units, typically
\( 0.0821\text{ L atm/mol K} \)
for pressure in atmospheres and volume in liters.

To solve for pressure, the Ideal Gas Law equation can be rearranged to:
\( P = \frac{nRT}{V} \).
This formula allows us to calculate the final pressure once volume, temperature, and the number of moles are known or have been determined.

Temperature Conversion to Kelvin

Temperature is a key variable in the ideal gas law and must be expressed in Kelvin, which is the SI unit of thermodynamic temperature. The Kelvin scale is an absolute temperature scale that starts at absolute zero, the point at which there is no thermal energy.

Converting Celsius to Kelvin

The formula for converting Celsius to Kelvin is simple:
\( T(K) = T(^\textdegree C) + 273.15 \),
where \( T(K) \) represents the temperature in Kelvin and \( T(^\textdegree C) \) is the temperature in Celsius. This conversion is critical for gas law problems, as using temperatures in Celsius can lead to incorrect results.

Remember that a change in temperature will affect the pressure and volume of a gas, as per Gay-Lussac’s Law and Charles’s Law, respectively. By always converting to Kelvin before using the ideal gas law, we ensure our calculations accurately reflect these temperature-dependent behaviors.

Molar Volume of Gases

The molar volume of a gas refers to the volume occupied by one mole of a gas at specified conditions of temperature and pressure. When dealing with gases under standard temperature and pressure (STP), which is defined as \(0^\textdegree C\) (or \(273.15 K\)) and \(1 atm\), the molar volume of an ideal gas is approximately
\(22.4 L/mol\).

Significance in Calculations

In gas calculations, understanding molar volume helps determine the amount of gas or number of moles during a transfer or reaction. As the molar volume can change with temperature or pressure, the conditions must be accounted for accurately.
For example, in non-STP conditions, you would utilize the Ideal Gas Law to determine the molar volume by solving for \(V/n\), which yields the volume per mole at the specific condition. This is crucial when gases are transferred from one container to another under different temperatures or pressures, as knowing the molar volume allows us to adjust the calculations to maintain accuracy in determining gas properties following the transfer.