The resistance of a coil used in a platinum-resistance thermometer at `0^(@)C` is `3.00Omega` and at `100^(@)C` is 3.75 2. Its resistance at an unknown temperature is measured as 3.15W. Calculate the unknown temperature.

Answer

To solve the problem, we will use the relationship between resistance and temperature for a platinum resistance thermometer. The formula is given by:

\[ R = R_0 (1 + \alpha \Delta T) \]

Where:
- \( R \) is the resistance at temperature \( T \)
- \( R_0 \) is the resistance at 0°C
- \( \alpha \) is the temperature coefficient of resistance
- \( \Delta T \) is the change in temperature from 0°C

### Step 1: Identify the known values
From the problem, we have:
- \( R_0 = 3.00 \, \Omega \) (resistance at 0°C)
- \( R_{100} = 3.75 \, \Omega \) (resistance at 100°C)
- \( R_{unknown} = 3.15 \, \Omega \) (resistance at unknown temperature)

### Step 2: Calculate the temperature coefficient \( \alpha \)
Using the resistance at 100°C, we can find \( \alpha \) using the formula:

\[ R_{100} = R_0 (1 + \alpha \cdot 100) \]

Substituting the known values:

\[ 3.75 = 3.00 (1 + \alpha \cdot 100) \]

Dividing both sides by 3.00:

\[ 1.25 = 1 + \alpha \cdot 100 \]

Now, subtract 1 from both sides:

\[ 0.25 = \alpha \cdot 100 \]

Now, divide by 100 to find \( \alpha \):

\[ \alpha = \frac{0.25}{100} = 0.0025 \, \text{°C}^{-1} \]

### Step 3: Use the value of \( \alpha \) to find the unknown temperature
Now we can use the resistance at the unknown temperature:

\[ R_{unknown} = R_0 (1 + \alpha \Delta T) \]

Substituting the known values:

\[ 3.15 = 3.00 (1 + 0.0025 \Delta T) \]

Dividing both sides by 3.00:

\[ 1.05 = 1 + 0.0025 \Delta T \]

Subtracting 1 from both sides:

\[ 0.05 = 0.0025 \Delta T \]

Now, divide by 0.0025 to find \( \Delta T \):

\[ \Delta T = \frac{0.05}{0.0025} = 20 \, \text{°C} \]

### Step 4: Calculate the unknown temperature
Since \( \Delta T \) is the change in temperature from 0°C:

\[ T = 0 + \Delta T = 20 \, \text{°C} \]

### Final Answer
The unknown temperature is \( 20 \, \text{°C} \).

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