The coeefficient of volume expansion of liquid is `gamma`. The fractional change in its density for `DeltaT` rise in tempeature 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 ΔT, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Coefficient of Volume Expansion**:
The coefficient of volume expansion (γ) describes how the volume of a substance changes with temperature. The relationship can be expressed as:
\[ V = V_0 (1 + \gamma \Delta T) \]
where \( V_0 \) is the initial volume and \( \Delta T \) is the change in temperature.

2. **Relate Density to Volume**:
Density (ρ) is defined as mass (m) divided by volume (V):
\[ \rho = \frac{m}{V} \]
If the mass of the liquid remains constant during the temperature change, any change in volume will affect the density.

3. **Express the New Density**:
After the volume expands due to the temperature increase, the new density (ρ) can be expressed as:
\[ \rho = \frac{m}{V} = \frac{m}{V_0 (1 + \gamma \Delta T)} \]

4. **Calculate the Initial Density**:
The initial density (ρ₀) before the temperature change is:
\[ \rho_0 = \frac{m}{V_0} \]

5. **Substitute for Volume in Density Equation**:
Now, substituting the expression for volume into the density equation gives:
\[ \rho = \frac{\rho_0 V_0}{V_0 (1 + \gamma \Delta T)} = \frac{\rho_0}{1 + \gamma \Delta T} \]

6. **Find the Change in Density**:
The change in density (Δρ) can be calculated as:
\[ \Delta \rho = \rho - \rho_0 = \frac{\rho_0}{1 + \gamma \Delta T} - \rho_0 \]
To simplify this, we can write:
\[ \Delta \rho = \rho_0 \left( \frac{1}{1 + \gamma \Delta T} - 1 \right) \]
\[ = \rho_0 \left( \frac{1 - (1 + \gamma \Delta T)}{1 + \gamma \Delta T} \right) \]
\[ = \rho_0 \left( \frac{-\gamma \Delta T}{1 + \gamma \Delta T} \right) \]

7. **Calculate the Fractional Change in Density**:
The fractional change in density is given by:
\[ \frac{\Delta \rho}{\rho_0} = \frac{-\gamma \Delta T}{1 + \gamma \Delta T} \]
For small values of \( \gamma \Delta T \), we can approximate:
\[ \frac{\Delta \rho}{\rho_0} \approx -\gamma \Delta T \]

### Final Answer:
The fractional change in density for a ΔT rise in temperature is approximately:
\[ \frac{\Delta \rho}{\rho_0} \approx -\gamma \Delta T \]