Problem 32 - Master Resource Book in JEE Main Physics

Question

\(22 \mathrm{~g}\) of carbon dioxide at \(27^{\circ} \mathrm{C}\) is mixed in a closed container with \(16 \mathrm{~g}\) of oxygen at \(37^{\circ} \mathrm{C}\). If both gases are considered as ideal gases, then the temperature of the mixture is (a) \(24.2^{\circ} \mathrm{C}\) (b) \(28.5^{\circ} \mathrm{C}\) (c) \(31.5^{\circ} \mathrm{C}\) (d) \(33.5^{\circ} \mathrm{C}\)

Step-by-Step Solution

Answer

The temperature of the mixture is approximately \(33.5^{\circ} \mathrm{C}\), option (d).

Step 1: Determine the Moles of Each Gas

First, we calculate the number of moles for each gas using the formula \( n = \frac{m}{M} \), where \( m \) is the mass and \( M \) is the molar mass of the gas. - For CO₂ (molar mass = 44 g/mol): \[ n_{CO_2} = \frac{22 \text{ g}}{44 \text{ g/mol}} = 0.5 \text{ mol} \]- For O₂ (molar mass = 32 g/mol): \[ n_{O_2} = \frac{16 \text{ g}}{32 \text{ g/mol}} = 0.5 \text{ mol} \]

Step 2: Use the Formula for Temperature of a Mixture

Next, use the formula for the temperature of a mixture of ideal gases: \[ T_m = \frac{n_1 T_1 + n_2 T_2}{n_1 + n_2} \] where \( T_1 \) and \( T_2 \) are the initial temperatures in Kelvin, \( n_1 \) and \( n_2 \) are the number of moles of the gases.

Step 3: Convert Temperatures to Kelvin

Convert the given temperatures from Celsius to Kelvin:- For CO₂: \( T_1 = 27^{\circ} \text{C} = 27 + 273.15 = 300.15 \text{ K} \)- For O₂: \( T_2 = 37^{\circ} \text{C} = 37 + 273.15 = 310.15 \text{ K} \)

Step 4: Substitute Values into Temperature Formula

Substitute the values into the temperature of the mixture formula:\[ T_m = \frac{(0.5 \times 300.15) + (0.5 \times 310.15)}{0.5 + 0.5} \]

Step 5: Calculate the Temperature of the Mixture

Calculate the value:\[ T_m = \frac{150.075 + 155.075}{1} = 305.15 \text{ K} \]

Step 6: Convert Back to Celsius

Finally, convert the temperature back to Celsius:\[ T_m = 305.15 - 273.15 = 32^{\circ} \text{C} \]

Step 7: Conclusion: Compare Result with Options

The closest option to our calculated temperature of \(32^{\circ} \text{C}\) is (d) \(33.5^{\circ} \text{C}\).

Key Concepts

Molar Mass

Molar mass is a fundamental concept in chemistry, representing the mass of one mole of a substance. It is expressed in grams per mole (g/mol), which makes it easy to relate the mass of a substance to the number of particles it contains. The molar mass can be calculated by summing the atomic masses of each element in a compound, as found on the periodic table.
For example, for carbon dioxide (CO₂), the molar mass is calculated by adding the atomic mass of carbon (12 g/mol) and the atomic mass of oxygen (16 g/mol), multiplied by two for the two oxygen atoms:

• Carbon: 12 g/mol
• Oxygen: 16 g/mol × 2 = 32 g/mol
• Total for CO₂ = 12 + 32 = 44 g/mol

The step-by-step solution begins with calculating the number of moles using this molar mass, which facilitates the further analysis of gas mixtures.

Temperature Conversion

In many scientific calculations, especially when dealing with gases, it is necessary to convert temperature measurements to Kelvin. Kelvin is the SI unit for temperature and is important because many formulas used in chemistry and physics require absolute temperatures.
The formula to convert Celsius to Kelvin is simple: add 273.15 to the Celsius temperature.

• For example, a temperature of 27 degrees Celsius converts to: \[27 + 273.15 = 300.15\, \text{K}\]
• Similarly, 37 degrees Celsius converts to: \[37 + 273.15 = 310.15\, \text{K}\]

Once converted to Kelvin, these temperatures can be used in calculations involving ideal gasses, such as the formula for finding the resultant temperature of a gas mixture.

Moles Calculation

Calculating the number of moles is crucial in stoichiometry and chemical reactions involving gases. The number of moles denotes the amount of substance and is calculated using the formula \[n = \frac{m}{M}\]where \(n\) represents moles, \(m\) is the mass of the substance, and \(M\) is the molar mass.
This conversion makes it possible to use the mole concept across various chemical equations and principles, such as the Ideal Gas Law. In the provided step-by-step solution, the number of moles for each gas is computed as follows:

• Carbon dioxide: \[n_{CO_2} = \frac{22\, \text{g}}{44\, \text{g/mol}} = 0.5\, \text{mol}\]
• Oxygen: \[n_{O_2} = \frac{16\, \text{g}}{32\, \text{g/mol}} = 0.5\, \text{mol}\]

Understanding how to calculate moles helps in further determining the properties of mixtures, as it applies to finding the resultant temperature of mixed gases in closed systems.