The value of coefficient of volume expansion of glycerin is `5 xx 10^(-4) K^(-1)`. The fractional change in the density of glycerin for a rise of `40^(@)C` in its temperature is

Options

A. `0.010`
B. `0.015`
C. `0.020`
D. `0.025`

Explanation

To find the fractional change in the density of glycerin due to a rise in temperature, we can follow these steps:

### Step 1: Understand the relationship between volume expansion and density
The coefficient of volume expansion (γ) relates to the change in volume (ΔV) of a substance with a change in temperature (ΔT). The formula for fractional change in volume is given by:
\[ \frac{\Delta V}{V} = \gamma \Delta T \]

### Step 2: Relate density to volume
Density (ρ) is defined as mass (m) divided by volume (V):
\[ \rho = \frac{m}{V} \]
When the temperature increases, the volume changes, which in turn affects the density.

### Step 3: Calculate the change in volume
Given:
- Coefficient of volume expansion, γ = \(5 \times 10^{-4} \, K^{-1}\)
- Change in temperature, ΔT = \(40 \, °C\)

Using the formula for fractional change in volume:
\[ \frac{\Delta V}{V} = \gamma \Delta T = (5 \times 10^{-4}) \times 40 = 2 \times 10^{-2} \]

### Step 4: Relate the change in volume to change in density
The fractional change in density (Δρ/ρ) can be related to the change in volume as follows:
\[ \frac{\Delta \rho}{\rho} = -\frac{\Delta V}{V + \Delta V} \]
For small changes, we can approximate:
\[ \frac{\Delta \rho}{\rho} \approx -\frac{\Delta V}{V} \]

Thus:
\[ \frac{\Delta \rho}{\rho} \approx -2 \times 10^{-2} \]

### Step 5: Conclusion
The fractional change in the density of glycerin for a rise of \(40 \, °C\) in its temperature is approximately:
\[ \frac{\Delta \rho}{\rho} \approx -0.02 \]