For a certain gas the ratio of specific heat is given to be `gamma=1.5` for this gas

Options

A. `C_(V)=3R`
B. `C_(p)=3R`
C. `C_(p)=5R`
D. `C_(V)=5R`

Explanation

To solve the problem regarding the specific heat ratio (gamma) for a gas, we will follow these steps:

### Step 1: Understand the Definition of Gamma
The ratio of specific heats, denoted as gamma (γ), is defined as:
\[ \gamma = \frac{C_p}{C_v} \]
where \(C_p\) is the specific heat at constant pressure and \(C_v\) is the specific heat at constant volume.

### Step 2: Given Value of Gamma
From the problem statement, we know:
\[ \gamma = 1.5 \]

### Step 3: Relate Cp and Cv
Using the definition of gamma, we can express \(C_p\) in terms of \(C_v\):
\[ C_p = \gamma \cdot C_v \]
Substituting the value of gamma:
\[ C_p = 1.5 \cdot C_v \]

### Step 4: Use the Relationship Between Cp, Cv, and R
We also know the relationship between \(C_p\), \(C_v\), and the gas constant \(R\):
\[ C_p - C_v = R \]
Substituting \(C_p\) from the previous step into this equation:
\[ 1.5 \cdot C_v - C_v = R \]

### Step 5: Simplify the Equation
Now, simplify the left side:
\[ (1.5 - 1) C_v = R \]
\[ 0.5 C_v = R \]

### Step 6: Solve for Cv
Now, we can solve for \(C_v\):
\[ C_v = \frac{R}{0.5} = 2R \]

### Step 7: Solve for Cp
Now that we have \(C_v\), we can find \(C_p\):
\[ C_p = 1.5 \cdot C_v = 1.5 \cdot 2R = 3R \]

### Final Result
Thus, the specific heat at constant pressure for the gas is:
\[ C_p = 3R \]

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