To find the specific heat of mercury, we can use the formula for calculating the heat absorbed or released by a substance, which is given by:
\[ Q = mc\Delta T \]
where:
- \( Q \) is the heat energy (in joules),
- \( m \) is the mass of the substance (in kilograms),
- \( c \) is the specific heat capacity (in J/(kg·K)),
- \( \Delta T \) is the change in temperature (in Kelvin).
From the problem, we know:
- \( Q = 257 , \text{J} \)
- \( m = 0.450 , \text{kg} \)
- \( \Delta T = 4.09 , \text{K} \)
We need to find the specific heat \( c \). Rearranging the formula gives:
\[ c = \frac{Q}{m \Delta T} \]
Now, substituting the given values into the equation:
\[ c = \frac{257 , \text{J}}{0.450 , \text{kg} \times 4.09 , \text{K}} \]
Calculating the denominator:
\[ 0.450 , \text{kg} \times 4.09 , \text{K} = 1.8435 , \text{kg·K} \]
Now, substituting this back into our equation for \( c \):
\[ c = \frac{257 , \text{J}}{1.8435 , \text{kg·K}} \approx 139.5 , \text{J/(kg·K)} \]
Thus, the specific heat of mercury is approximately:
\[ \boxed{139.5 , \text{J/(kg·K)}} \]