Problem 38 - Master Resource Book in JEE Main Physics

Question

Two moles of monoatomic gas is mixed with three moles of a diatomic gas. The molar specific heat of the mixture at constant volume is (a) \(1.55 R\) (b) \(2.10 R\) (c) \(1.63 R\) (d) \(2.20 R\)

Step-by-Step Solution

Answer

(b) The molar specific heat of the mixture at constant volume is \(2.10 R\).

Step 1: Determine the Specific Heat Capacities

A monoatomic gas has a molar specific heat at constant volume ( \(C_{V, ext{mono}} \)) given by \(\frac{3}{2} R\). A diatomic gas has molar specific heat at constant volume ( \(C_{V, ext{di}} \)) given by \(\frac{5}{2} R\).

Step 2: Calculate Total Moles

Add the moles of both gases to find the total moles in the mixture: \(2 + 3 = 5\) moles.

Step 3: Calculate Contribution of Each Gas

Calculate the contribution to the heat capacity by each type of gas: \(2 \, \text{moles}\ \times \frac{3}{2} R = 3 R\) for the monoatomic gas and \(3 \, \text{moles}\ \times \frac{5}{2} R = 7.5 R\) for the diatomic gas.

Step 4: Calculate Total Heat Capacity

Sum the contributions from both gases to find the total heat capacity of the mixture: \(3 R + 7.5 R = 10.5 R\).

Step 5: Calculate Molar Specific Heat of Mixture

Divide the total heat capacity by the total number of moles to get the molar specific heat of the mixture: \(\frac{10.5 R}{5} = 2.10 R\).

Key Concepts

Molar Specific Heat Capacity

Molar specific heat capacity is an important concept in thermodynamics. It refers to the amount of heat needed to raise the temperature of one mole of a substance by one degree Celsius (or Kelvin). This concept plays a key role when working with gaseous mixtures because it allows us to calculate how different gases store and transfer energy.

For gases, the molar specific heat can be measured at constant volume or constant pressure, but when dealing with mixtures, we often consider the heat capacity at constant volume (\(C_V\)). This is because it simplifies calculations, especially in closed systems. The molar specific heat capacity helps us understand how much energy is required to influence temperature changes in substances, assisting in a variety of applications from industry to weather prediction.

Monoatomic Gas

Monoatomic gases are gases that consist of single atoms. These are often noble gases such as helium (\(He\)), neon (\(Ne\)), and argon (\(Ar\)). These gases have a specific heat capacity at constant volume of \(\frac{3}{2} R\), where \(R\) is the universal gas constant.

• It is notable because monoatomic gases have the lowest possible heat capacity for a gas due to their simple atomic structure.
• This simplicity means there are fewer degrees of freedom for energy absorption beyond translational movement.
• This makes them easier to analyze, especially when mixed with other, more complex gases.

In practical problems, knowing the specific heat capacity at constant volume allows scientists to predict how these gases will behave in different thermodynamic processes.

Diatomic Gas

Unlike monoatomic gases, diatomic gases consist of molecules made up of two atoms. Common examples include oxygen (\(O_2\)), nitrogen (\(N_2\)), and hydrogen (\(H_2\)). These gases exhibit a slightly higher molar specific heat capacity at constant volume, which is given by \(\frac{5}{2} R\).

• Diatomic gases have additional degrees of freedom compared to monoatomic gases. These include rotational motion in addition to translational movement.
• Because of this, they require more energy to increase their temperature by a given amount.
• In thermodynamic systems, this means diatomic gases will store and release energy differently than monoatomic gases.

Understanding the molar specific heat capacity of diatomic gases is crucial when dealing with mixtures, as it directly impacts the energy dynamics and overall heat capacity of the mixture.