The amount of heat required to raise the temperature of 10 g liquid from `0^(@)C` to `10^(@)C,` if its specific heat (in `(cal)/(g ^(@)C)`) varies with temperature as `0.3t^(2)` (where t is temperature in `""^(@)C`) is

Options

A. 2 kcal
B. 3 kcal
C. 2.5 kcal
D. 1 kcal

Explanation

To solve the problem of calculating the amount of heat required to raise the temperature of a 10 g liquid from \(0^\circ C\) to \(10^\circ C\) with a specific heat that varies with temperature as \(0.3t^2\), we can follow these steps:

### Step 1: Understand the Heat Equation
The heat \(q\) required to change the temperature of a substance can be calculated using the formula:
\[ q = m \int_{T_1}^{T_2} S(T) \, dT \]
where:
- \(m\) is the mass of the substance,
- \(S(T)\) is the specific heat capacity as a function of temperature,
- \(T_1\) and \(T_2\) are the initial and final temperatures.

### Step 2: Identify Given Values
From the problem, we have:
- Mass \(m = 10 \, \text{g}\)
- Specific heat \(S(t) = 0.3t^2\)
- Initial temperature \(T_1 = 0^\circ C\)
- Final temperature \(T_2 = 10^\circ C\)

### Step 3: Set Up the Integral
We need to calculate:
\[ q = 10 \int_{0}^{10} 0.3t^2 \, dt \]

### Step 4: Calculate the Integral
First, we can factor out the constant \(0.3\) and the mass \(10\):
\[ q = 10 \times 0.3 \int_{0}^{10} t^2 \, dt \]
Now, we need to compute the integral \(\int t^2 \, dt\):
\[ \int t^2 \, dt = \frac{t^3}{3} \]
Evaluating this from \(0\) to \(10\):
\[ \int_{0}^{10} t^2 \, dt = \left[\frac{t^3}{3}\right]_{0}^{10} = \frac{10^3}{3} - \frac{0^3}{3} = \frac{1000}{3} \]

### Step 5: Substitute Back into the Heat Equation
Now substituting back into the equation for \(q\):
\[ q = 10 \times 0.3 \times \frac{1000}{3} \]
Calculating this gives:
\[ q = 10 \times 0.3 \times \frac{1000}{3} = 10 \times 0.3 \times 333.33 \approx 1000 \times 0.1 = 100 \, \text{calories} \]

### Step 6: Convert to Kilocalories
Since \(1 \, \text{kcal} = 1000 \, \text{cal}\):
\[ q = \frac{100}{1000} = 0.1 \, \text{kcal} \]

### Final Answer
The total amount of heat required to raise the temperature of the liquid from \(0^\circ C\) to \(10^\circ C\) is:
\[ \boxed{1 \, \text{kcal}} \]