A 2.00 kg metal object requires `5.02 xx 10^(3) J` of heat to raise its temperature from `20.0^(@)C` to `40.0^(@)C`. What is the specific heat capacity of the metal ?

Options

A. `63.0J //( kg. ""^(@)C )`
B. `126J //(kg. ""^(@)C )`
C. `251 J //( kg. ""^(@)C )`
D. `502 J // ( kg . ""^(@ )C)`

Answer

To find the specific heat capacity of the metal object, we can use the formula for heat transfer:

\[ Q = m \cdot S \cdot \Delta T \]

Where:
- \( Q \) is the heat energy absorbed (in joules),
- \( m \) is the mass of the object (in kilograms),
- \( S \) is the specific heat capacity (in joules per kilogram per degree Celsius),
- \( \Delta T \) is the change in temperature (in degrees Celsius).

### Step 1: Identify the known values
- Mass of the metal object, \( m = 2.00 \, \text{kg} \)
- Heat energy absorbed, \( Q = 5.02 \times 10^3 \, \text{J} \)
- Initial temperature, \( T_i = 20.0 \, \text{°C} \)
- Final temperature, \( T_f = 40.0 \, \text{°C} \)

### Step 2: Calculate the change in temperature
\[ \Delta T = T_f - T_i = 40.0 \, \text{°C} - 20.0 \, \text{°C} = 20.0 \, \text{°C} \]

### Step 3: Substitute the known values into the heat transfer formula
\[ Q = m \cdot S \cdot \Delta T \]
Substituting the values we have:
\[ 5.02 \times 10^3 \, \text{J} = 2.00 \, \text{kg} \cdot S \cdot 20.0 \, \text{°C} \]

### Step 4: Rearrange the equation to solve for specific heat capacity \( S \)
\[ S = \frac{Q}{m \cdot \Delta T} \]
Substituting the values:
\[ S = \frac{5.02 \times 10^3 \, \text{J}}{2.00 \, \text{kg} \cdot 20.0 \, \text{°C}} \]

### Step 5: Calculate \( S \)
\[ S = \frac{5.02 \times 10^3 \, \text{J}}{40.0 \, \text{kg°C}} = \frac{5020 \, \text{J}}{40.0 \, \text{kg°C}} = 125.5 \, \text{J/kg°C} \]

### Step 6: Final answer
The specific heat capacity of the metal is approximately:
\[ S \approx 126 \, \text{J/kg°C} \]