Question

An open vessel at \(27^{\circ} \mathrm{C}\) is heated until \(3 / 4\) of the air in it has been expelled. Assuming that the volume of the vessel remains constant, the temperature to which the vessel has been heated (1) \(210^{\circ} \mathrm{C}\) (2) \(1200^{\circ} \mathrm{C}\) (3) \(220^{\circ} \mathrm{C}\) (4) \(927^{\circ} \mathrm{C}\).

Short answer

927^{\circ}\mathrm{C}, Option (4).

Step-by-step solution

- Understand the Problem

The problem requires finding the temperature to which a vessel must be heated in order to expel three-fourths of the air initially present, assuming the volume of the vessel remains constant.

- Use the Ideal Gas Law

Since the volume is constant and the amount of expelled air is known, use the ideal gas law in the form of \(P_1/T_1 = P_2/T_2\). Here, \(P_1\) and \(T_1\) are the initial pressure and temperature, and \(P_2\) and \(T_2\) are the final pressure and temperature.

- Establish Initial and Final Conditions

Initially, the temperature \(T_1\) is \(27^{\circ}\mathrm{C} = 300 \,\mathrm{K}\). Since three-fourths of the air is expelled, only one-fourth remains, which means the final pressure \(P_2 = \frac{1}{4}P_1\).

- Solve for Final Temperature

Substitute the known values into the relationship: \(\frac{P_1}{300 \,\mathrm{K}} = \frac{\frac{1}{4}P_1}{T_2}\). Simplify to \(\frac{1}{300} = \frac{1}{4T_2}\). Solving for \(T_2\) gives \(T_2 = 4 \times 300 = 1200\,\mathrm{K}\).

- Convert to Celsius

Convert the final temperature from Kelvin to Celsius: \(T_2^{\circ}\mathrm{C} = 1200 \,\mathrm{K} - 273 = 927^{\circ}\mathrm{C}\).

- Select the Correct Option

The final temperature in Celsius is \(927^{\circ}\mathrm{C}\), which corresponds to option (4).

Key concepts

Temperature Calculation

When solving problems related to temperature changes in an ideal gas, it is essential to understand how temperature plays a role in the behavior of gases. Temperature is a measure of the average kinetic energy of gas molecules. In this exercise, the initial temperature of the vessel is given as 27°C. To computationally engage with the Ideal Gas Law effectively, we need to use the Kelvin scale. Convert Celsius to Kelvin by adding 273 to the given Celsius temperature: 27 + 273 = 300 K. This initial temperature will then be used in subsequent calculations to determine how much the temperature must increase to achieve the desired effect in the problem scenario.

Pressure and Volume Relationship

In the Ideal Gas Law, pressure, volume, and temperature are interconnected. The relationship is given by \(PV = nRT\), which can be rearranged to show how these variables interact at constant volume. For this exercise, the volume of the vessel remains unchanged. As the gas is heated, the pressure must balance the temperature change. Initially, let's denote the initial pressure as P₁ and initial temperature as T₁ = 300 K. When three-fourths of the air is expelled, only one-fourth of the initial air remains, making the final pressure P₂ equal to one-fourth of P₁. Using \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \), substitute the given values and solve for T₂, leading to \( T_2 = 4 \times 300 = 1200 \) K.

Kelvin to Celsius Conversion

Once the final temperature in Kelvin is calculated, often the next step is converting this value to Celsius. This is particularly useful as temperature values in degrees Celsius are more common in practical situations. The conversion formula between Kelvin and Celsius is straightforward: subtract 273 from the Kelvin temperature. So, for the final temperature calculated in the exercise, which is 1200 K, we convert it to Celsius by computing 1200 - 273 = 927°C. Therefore, the temperature to which the vessel has been heated is 927°C.