To find the specific heat capacity of the metal, we can use the formula for specific heat capacity ([tex]\(c\)[/tex]):
[tex]\[ c = \frac{Q}{m \Delta T} \][/tex]
where:
- [tex]\( Q \)[/tex] is the amount of heat absorbed (in Joules, [tex]\(J\)[/tex]),
- [tex]\( m \)[/tex] is the mass of the metal (in grams, [tex]\(g\)[/tex]),
- [tex]\(\Delta T\)[/tex] is the temperature change (in degrees Celsius, [tex]\(°C\)[/tex]).
Given the values:
- [tex]\( Q = 218.17 \, J \)[/tex]
- [tex]\( m = 14.407 \, g \)[/tex]
- [tex]\(\Delta T = 37.9 \, °C\)[/tex]
we can substitute these values into the formula to find [tex]\( c \)[/tex]:
[tex]\[ c = \frac{218.17 \, J}{14.407 \, g \times 37.9 \, °C} \][/tex]
First, we carry out the multiplication in the denominator:
[tex]\[ 14.407 \, g \times 37.9 \, °C = 546.0453 \, g \cdot °C \][/tex]
Now, we divide the numerator by this product:
[tex]\[ c = \frac{218.17 \, J}{546.0453 \, g \cdot °C} \][/tex]
[tex]\[ c \approx 0.39956024015736996 \, \frac{J}{g \cdot °C} \][/tex]
Therefore, the specific heat capacity of the metal is approximately [tex]\(0.400 \, \frac{J}{g \cdot °C}\)[/tex] (rounded to three decimal places).
