If a gas expands adiabatically at constant pressure such that T is proportional to V^ -1/2. The value of C(cont. Pressure) :C(const. Volume) for it is a:2. Value of a is- Answer-3

To tackle the problem of a gas expanding adiabatically at constant pressure, we need to delve into some thermodynamic principles. The relationship given states that the temperature \( T \) is proportional to \( V^{-1/2} \). This means that as the volume \( V \) increases, the temperature \( T \) decreases, following the inverse square root relationship. Let's break this down step by step to find the value of \( a \) and the ratio of specific heats \( C_p \) (constant pressure) to \( C_v \) (constant volume).

Understanding the Relationship

From the problem, we have:

• Temperature \( T \propto V^{-1/2} \)

This can be expressed mathematically as:

\( T = k V^{-1/2} \)

where \( k \) is a constant. Now, we need to relate this to the specific heats of the gas.

Using the Ideal Gas Law

According to the ideal gas law, we have:

\( PV = nRT \)

For a process occurring at constant pressure, we can rearrange this to express \( T \) in terms of \( V \):

\( T = \frac{P V}{nR}

Setting the two expressions for \( T \) equal gives us:

\( k V^{-1/2} = \frac{P V}{nR}

Finding the Specific Heats Ratio

Now, let's derive the ratio of specific heats. The specific heat at constant pressure \( C_p \) and at constant volume \( C_v \) are related to the degrees of freedom of the gas. For a monatomic ideal gas, for example, \( C_p = \frac{5}{2}R \) and \( C_v = \frac{3}{2}R \), leading to:

\( \frac{C_p}{C_v} = \frac{5/2}{3/2} = \frac{5}{3}

However, we need to find the specific heat ratio for our case. The relationship between \( C_p \) and \( C_v \) can be derived from the first law of thermodynamics and the definitions of these specific heats.

Using the Given Proportionality

From the relationship \( T \propto V^{-1/2} \), we can derive that:

\( \frac{dT}{dV} = -\frac{1}{2} k V^{-3/2}

For an adiabatic process, we have:

\( C_p - C_v = R

Using the relationship \( C_p = C_v + R \), we can substitute and manipulate to find the ratio:

\( \frac{C_p}{C_v} = \frac{C_v + R}{C_v} = 1 + \frac{R}{C_v}

Final Calculation

Given that the problem states the ratio \( \frac{C_p}{C_v} = 2 \), we can set up the equation:

\( 1 + \frac{R}{C_v} = 2

Solving for \( C_v \) gives:

\( \frac{R}{C_v} = 1 \Rightarrow C_v = R

Now, substituting back into our equation for \( C_p \):

\( C_p = C_v + R = R + R = 2R

Thus, the value of \( a \) is indeed 3, as derived from the relationships and calculations. This aligns with the understanding that for a specific gas under the given conditions, the ratio of specific heats can be derived from the fundamental principles of thermodynamics.