If the temperature of body is increased by 100°C then the percentage decrease in it's density is [if `gamma` =`75 ×10^-5°C^-1`]

Options

A. 0.75
B. 0.075
C. 0.0075
D. 0.0008

Answer

To solve the problem of finding the percentage decrease in density when the temperature of a body is increased by 100°C, we will follow these steps:

### Step 1: Understand the relationship between temperature, volume, and density
When the temperature of a body increases, its volume also increases due to thermal expansion. The relationship between volume change and temperature change can be expressed as:
\[ V = V_0 (1 + \gamma \Delta T) \]
where:
- \( V \) = new volume
- \( V_0 \) = original volume
- \( \gamma \) = coefficient of volume expansion
- \( \Delta T \) = change in temperature

### Step 2: Substitute the known values
Given:
- \( \Delta T = 100°C \)
- \( \gamma = 75 \times 10^{-5} °C^{-1} \)

Substituting these values into the volume expansion formula:
\[ V = V_0 \left(1 + 75 \times 10^{-5} \times 100\right) \]
\[ V = V_0 \left(1 + 0.075\right) \]
\[ V = V_0 \times 1.075 \]

### Step 3: Relate density to volume
Density (\( \rho \)) is defined as mass (\( m \)) divided by volume (\( V \)):
\[ \rho = \frac{m}{V} \]
Thus, the new density (\( \rho' \)) after the temperature increase can be expressed as:
\[ \rho' = \frac{m}{V} = \frac{m}{V_0 \times 1.075} \]

### Step 4: Calculate the original density
The original density (\( \rho_0 \)) is:
\[ \rho_0 = \frac{m}{V_0} \]

### Step 5: Find the new density in terms of the original density
Now we can express the new density in terms of the original density:
\[ \rho' = \frac{\rho_0}{1.075} \]

### Step 6: Calculate the change in density
The change in density (\( \Delta \rho \)) can be calculated as:
\[ \Delta \rho = \rho_0 - \rho' = \rho_0 - \frac{\rho_0}{1.075} \]
\[ \Delta \rho = \rho_0 \left(1 - \frac{1}{1.075}\right) \]
\[ \Delta \rho = \rho_0 \left(\frac{1.075 - 1}{1.075}\right) \]
\[ \Delta \rho = \rho_0 \left(\frac{0.075}{1.075}\right) \]

### Step 7: Calculate the percentage decrease in density
The percentage decrease in density is given by:
\[ \text{Percentage decrease} = \left(\frac{\Delta \rho}{\rho_0}\right) \times 100 \]
Substituting the expression for \( \Delta \rho \):
\[ \text{Percentage decrease} = \left(\frac{\rho_0 \left(\frac{0.075}{1.075}\right)}{\rho_0}\right) \times 100 \]
\[ \text{Percentage decrease} = \left(\frac{0.075}{1.075}\right) \times 100 \]

### Step 8: Calculate the final value
Calculating the numerical value:
\[ \text{Percentage decrease} \approx \left(0.069767\right) \times 100 \approx 6.98\% \]

Thus, the percentage decrease in density when the temperature of the body is increased by 100°C is approximately **6.98%**.

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