The coefficient of volume expansion of liquid is `gamma`. The fractional change in its density for `DeltaT` rise in temperature is

Options

A. `gammaDeltaT`
B. `(DeltaT)/(gamma)`
C. `1+ gamma Deltat`
D. `1- gamma DeltaT`

Answer

To find the fractional change in density of a liquid when the temperature rises by \( \Delta T \), we can follow these steps:

### Step 1: Understand the relationship between volume and temperature
The volume expansion of a liquid due to a temperature change is described by the formula:
\[ V = V_0 (1 + \gamma \Delta T) \]
where:
- \( V \) is the final volume,
- \( V_0 \) is the initial volume,
- \( \gamma \) is the coefficient of volume expansion,
- \( \Delta T \) is the change in temperature.

### Step 2: Relate density to volume
Density (\( d \)) is defined as mass (\( m \)) divided by volume (\( V \)):
\[ d = \frac{m}{V} \]
Since mass remains constant during the temperature change, we can express the initial density (\( d_0 \)) and the final density (\( d \)) as follows:
\[ d_0 = \frac{m}{V_0} \]
\[ d = \frac{m}{V} \]

### Step 3: Substitute the volume expansion into the density formula
Using the volume expansion formula, we can express the final density in terms of the initial density:
\[ d = \frac{m}{V_0 (1 + \gamma \Delta T)} = \frac{d_0 V_0}{V_0 (1 + \gamma \Delta T)} = \frac{d_0}{1 + \gamma \Delta T} \]

### Step 4: Calculate the change in density
The change in density (\( \Delta d \)) can be expressed as:
\[ \Delta d = d - d_0 = \frac{d_0}{1 + \gamma \Delta T} - d_0 \]
To combine these terms, we can find a common denominator:
\[ \Delta d = d_0 \left( \frac{1}{1 + \gamma \Delta T} - 1 \right) = d_0 \left( \frac{1 - (1 + \gamma \Delta T)}{1 + \gamma \Delta T} \right) = d_0 \left( \frac{-\gamma \Delta T}{1 + \gamma \Delta T} \right) \]

### Step 5: Find the fractional change in density
The fractional change in density is given by:
\[ \text{Fractional change} = \frac{\Delta d}{d_0} = \frac{-\gamma \Delta T}{1 + \gamma \Delta T} \]
For small values of \( \gamma \Delta T \), we can approximate \( 1 + \gamma \Delta T \approx 1 \), leading to:
\[ \text{Fractional change} \approx -\gamma \Delta T \]

### Conclusion
Thus, the fractional change in density for a rise in temperature \( \Delta T \) is approximately:
\[ \text{Fractional change in density} \approx -\gamma \Delta T \]

### Final Answer
The correct option is:
\[ \text{Option D: } 1 - \gamma \Delta T \]