Question

A body cools from a temperature $3T$ to $2T$ in $10\, minutes$. The room temperature is $T$. Assume that Newton�s law of cooling is applicable. The temperature of the body at the end of next $10\,minutes$ will be -

• A. $\frac{7}{4} T$
• B. $\frac{3}{2} T$
• C. $\frac{4}{3} T$
• D. $T$

Answer 1

The Correct Option is B

Thermodynamics Calculation: Finding \( T_f \)

We are given a thermodynamic process involving temperature changes. Let’s break down the problem step by step:

Step 1: Temperature Relationships 

We start with the following temperature relationships:

• \(3T \rightarrow [t_1 = 10 \, \text{min}]\)
• \(2T \rightarrow [t_2 = 10 \, \text{min}]\)
• \(T_f\  is\ the\ final\ temperature\)

Step 2: Formulating the First Equation

We are given the temperature changes over a time interval and can use the following relationship between the initial and final temperatures:

\(T_0 = T\)

The first equation is based on the temperature change from \( 3T \) to \( 2T \) over 10 minutes:

\(\left(\frac{3T - 2T}{10}\right) = c_1 \left(\frac{3T + 2T}{2} - T\right) \) ....(\)

This equation is based on the fact that heat exchange rate is proportional to the temperature difference.

Step 3: Formulating the Second Equation

The second equation is based on the temperature change from \( 2T \) to \( T_f \) over 10 minutes:

\(\left(\frac{2T - T_f}{10}\right) = c_1 \left(\frac{2T + T_f}{2} - T\right) \) ....(i\)

Step 4: Solving the Equations

Next, we need to take the ratio of the two equations (i) and (ii) to solve for the final temperature \( T_f \):

\(\frac{E(i)}{E(ii)} \Rightarrow \frac{\frac{T}{10}}{\frac{2T - T_f}{10}} = \frac{\frac{5T - 2T}{2}}{\frac{T_f}{2}}\)

Step 5: Simplifying the Equation

After simplifying the equation, we get:

\(\frac{T}{2T - T_f} = \frac{3T}{T_f}\)

Step 6: Solving for \( T_f \)

Rearranging this equation to solve for \( T_f \), we get:

\(T_f = 6T - 3T_f\)

Now, simplify further:

\(4T_f = 6T\)

Finally, we find the value of \( T_f \):

\(T_f = \frac{3}{2} T\)

Conclusion:

The final temperature \( T_f \) is:

\(T_f = \frac{3}{2} T\)