A thermally insulated piece of metal is heated under atmospheric pressure by an electric current so that it receives electric energy at a constant power P. This leads to an increase of absolute temperature T of the metal with time t as follows:
`T(t)=T_0[1+a(t-t_0)]^(1//4)`. Here, a, `t_0` and `T_0` are constants. The heat capacity `C_p(T)` of the metal is

Options

A. `(4P)/(aT_(0))`
B. `(4PT^(3))/(aT_(0)^(4))`
C. `(2PT^(3))/(aT_(0)^(4))`
D. `(2P)/(aT_(0))`

Answer

To find the heat capacity \( C_p(T) \) of the metal, we start with the given temperature function:

\[ T(t) = T_0 \left[ 1 + a(t - t_0) \right]^{1/4} \]

### Step 1: Differentiate the temperature function

To find the heat capacity, we need to determine how the temperature changes with respect to time. We differentiate the temperature function \( T(t) \) with respect to time \( t \):

\[ \frac{dT}{dt} = \frac{d}{dt} \left( T_0 \left[ 1 + a(t - t_0) \right]^{1/4} \right) \]

Using the chain rule, we get:

\[ \frac{dT}{dt} = T_0 \cdot \frac{1}{4} \left[ 1 + a(t - t_0) \right]^{-3/4} \cdot a \]

This simplifies to:

\[ \frac{dT}{dt} = \frac{a T_0}{4} \left[ 1 + a(t - t_0) \right]^{-3/4} \]

### Step 2: Express the heat transfer in terms of power

The power \( P \) supplied to the metal is related to the heat transfer by:

\[ P = \frac{dQ}{dt} \]

Where \( dQ \) is the heat added. The heat capacity \( C_p \) is defined as:

\[ C_p = \frac{dQ}{dT} \]

### Step 3: Relate \( dQ \) to \( dT \)

From the definition of heat capacity, we can express \( dQ \) as:

\[ dQ = C_p \, dT \]

Substituting this into the power equation gives:

\[ P = C_p \frac{dT}{dt} \]

### Step 4: Rearranging for \( C_p \)

Rearranging the equation for \( C_p \) gives:

\[ C_p = \frac{P}{\frac{dT}{dt}} \]

### Step 5: Substitute \( \frac{dT}{dt} \)

Now we substitute our expression for \( \frac{dT}{dt} \):

\[ C_p = \frac{P}{\frac{a T_0}{4} \left[ 1 + a(t - t_0) \right]^{-3/4}} \]

This can be simplified to:

\[ C_p = \frac{4P \left[ 1 + a(t - t_0) \right]^{3/4}}{a T_0} \]

### Step 6: Express \( C_p \) in terms of \( T \)

We know from the original temperature equation that:

\[ 1 + a(t - t_0) = \left( \frac{T}{T_0} \right)^4 \]

Thus, substituting this back into our expression for \( C_p \):

\[ C_p = \frac{4P \left( \frac{T}{T_0} \right)^3}{a T_0} \]

### Final Result

Therefore, the heat capacity \( C_p(T) \) of the metal is:

\[ C_p(T) = \frac{4PT^3}{a T_0^4} \]