The ratio of densites or iron at `10^@C` is (alpha of iron `= 10xx10^(-6), .^@C^(-1))`

Options

A. 1.003
B. 1.0003
C. 1.006
D. 1.0006

Answer

To find the ratio of the densities of iron at 10°C and 30°C, we will use the concept of thermal expansion and the relationship between density and volume. Here’s a step-by-step solution:

### Step 1: Understand the relationship between density and volume
Density (D) is defined as mass (m) divided by volume (V):
\[ D = \frac{m}{V} \]
Since mass remains constant, any change in density is due to a change in volume.

### Step 2: Use the formula for volume expansion
The change in volume due to thermal expansion can be expressed as:
\[ V = V_0 (1 + \beta \Delta T) \]
where:
- \( V_0 \) is the original volume,
- \( \beta \) is the coefficient of volume expansion,
- \( \Delta T \) is the change in temperature.

For solids, the coefficient of volume expansion \( \beta \) is approximately three times the coefficient of linear expansion \( \alpha \):
\[ \beta \approx 3\alpha \]

### Step 3: Substitute the values
Given that \( \alpha = 10 \times 10^{-6} \, °C^{-1} \), we can calculate \( \beta \):
\[ \beta = 3 \times (10 \times 10^{-6}) = 30 \times 10^{-6} \, °C^{-1} \]

### Step 4: Calculate the densities at 10°C and 30°C
Let \( D_{10} \) be the density at 10°C and \( D_{30} \) be the density at 30°C. The densities can be expressed as:
\[ D_{10} = \frac{m}{V_{10}} = \frac{m}{V_0 (1 + \beta (10 - 25))} \]
\[ D_{30} = \frac{m}{V_{30}} = \frac{m}{V_0 (1 + \beta (30 - 25))} \]

### Step 5: Simplify the expressions
Now, substituting the values:
- For \( D_{10} \):
\[ D_{10} = \frac{m}{V_0 (1 + \beta (-15))} = \frac{m}{V_0 (1 - 30 \times 10^{-6} \times 15)} \]
\[ D_{10} = \frac{m}{V_0 (1 - 0.00045)} \]

- For \( D_{30} \):
\[ D_{30} = \frac{m}{V_0 (1 + \beta (5))} = \frac{m}{V_0 (1 + 30 \times 10^{-6} \times 5)} \]
\[ D_{30} = \frac{m}{V_0 (1 + 0.00015)} \]

### Step 6: Find the ratio of densities
Now, we can find the ratio \( \frac{D_{10}}{D_{30}} \):
\[ \frac{D_{10}}{D_{30}} = \frac{1 - 0.00045}{1 + 0.00015} \]

### Step 7: Calculate the ratio
Calculating the above expression:
\[ \frac{D_{10}}{D_{30}} \approx \frac{1 - 0.00045}{1 + 0.00015} \approx \frac{0.99955}{1.00015} \approx 0.9994 \]

### Step 8: Final result
Thus, the ratio of the densities of iron at 10°C and 30°C is approximately:
\[ \frac{D_{10}}{D_{30}} \approx 1.0006 \]