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

Answer

To solve the problem of finding the unknown temperature based on the resistance measurements of a platinum-resistance thermometer, we can follow these steps:

### Step 1: Understand the relationship between resistance and temperature
The resistance \( R_T \) at temperature \( T \) is given by the formula:
\[ R_T = R_0 \left(1 + \alpha (T - T_0)\right) \]
where:
- \( R_T \) = resistance at temperature \( T \)
- \( R_0 \) = resistance at reference temperature \( T_0 \)
- \( \alpha \) = temperature coefficient of resistance

### Step 2: Identify the known values
From the problem statement:
- At \( 0^\circ C \) (which is \( T_0 \)), \( R_0 = 3.00 \, \Omega \)
- At \( 100^\circ C \), \( R_{100} = 3.75 \, \Omega \)
- At an unknown temperature \( T \), \( R_T = 3.15 \, \Omega \)

### Step 3: Calculate the temperature coefficient \( \alpha \)
Using the resistance at \( 100^\circ C \):
\[ R_{100} = R_0 \left(1 + \alpha (100 - 0)\right) \]
Substituting the known values:
\[ 3.75 = 3.00 \left(1 + 100\alpha\right) \]
Dividing both sides by 3.00:
\[ 1.25 = 1 + 100\alpha \]
Subtracting 1 from both sides:
\[ 0.25 = 100\alpha \]
Solving for \( \alpha \):
\[ \alpha = \frac{0.25}{100} = 0.0025 \, \text{per } ^\circ C = 25 \times 10^{-4} \, \text{per } ^\circ C \]

### Step 4: Use the resistance formula to find the unknown temperature \( T \)
Now we can use the resistance at the unknown temperature:
\[ R_T = R_0 \left(1 + \alpha (T - T_0)\right) \]
Substituting the known values:
\[ 3.15 = 3.00 \left(1 + 25 \times 10^{-4} (T - 0)\right) \]
Dividing both sides by 3.00:
\[ 1.05 = 1 + 25 \times 10^{-4} T \]
Subtracting 1 from both sides:
\[ 0.05 = 25 \times 10^{-4} T \]
Solving for \( T \):
\[ T = \frac{0.05}{25 \times 10^{-4}} = \frac{0.05 \times 10^4}{25} = \frac{500}{25} = 20^\circ C \]

### Final Answer
The unknown temperature \( T \) is \( 20^\circ C \).

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