To determine the specific heat capacity of the unknown substance, we can use the formula:
[tex]\[ q = m C_p \Delta T \][/tex]
where:
- [tex]\( q \)[/tex] is the heat added, in joules (J)
- [tex]\( m \)[/tex] is the mass of the substance, in kilograms (kg)
- [tex]\( C_p \)[/tex] is the specific heat capacity, in J/(kg·°C)
- [tex]\( \Delta T \)[/tex] is the change in temperature, in degrees Celsius (°C)
Here are the given values:
- [tex]\( q = 3000.0 \, \text{J} \)[/tex]
- [tex]\( m = 0.465 \, \text{kg} \)[/tex]
- Initial temperature, [tex]\( T_{\text{initial}} = 50.0 \, ^{\circ} \text{C} \)[/tex]
- Final temperature, [tex]\( T_{\text{final}} = 100.0 \, ^{\circ} \text{C} \)[/tex]
1. Calculate the temperature change ([tex]\( \Delta T \)[/tex]):
[tex]\[ \Delta T = T_{\text{final}} - T_{\text{initial}} \][/tex]
[tex]\[ \Delta T = 100.0 \, ^{\circ} \text{C} - 50.0 \, ^{\circ} \text{C} \][/tex]
[tex]\[ \Delta T = 50.0 \, ^{\circ} \text{C} \][/tex]
2. Rearrange the formula to solve for [tex]\( C_p \)[/tex]:
[tex]\[ C_p = \frac{q}{m \Delta T} \][/tex]
3. Plug in the values and solve for [tex]\( C_p \)[/tex]:
[tex]\[ C_p = \frac{3000.0 \, \text{J}}{0.465 \, \text{kg} \times 50.0 \, ^{\circ} \text{C}} \][/tex]
4. Perform the multiplication in the denominator:
[tex]\[ 0.465 \, \text{kg} \times 50.0 \, ^{\circ} \text{C} = 23.25 \, \text{kg} \cdot ^{\circ} \text{C} \][/tex]
5. Now, compute [tex]\( C_p \)[/tex]:
[tex]\[ C_p = \frac{3000.0 \, \text{J}}{23.25 \, \text{kg} \cdot ^{\circ} \text{C}} \][/tex]
[tex]\[ C_p \approx 129.032 \, \text{J/(kg·°C)} \][/tex]
6. Convert the specific heat capacity to J/(g·°C):
Since [tex]\( 1 \, \text{kg} = 1000 \, \text{g} \)[/tex], divide by 1000:
[tex]\[ C_p = \frac{129.032 \, \text{J/(kg·°C)}}{1000} \][/tex]
[tex]\[ C_p \approx 0.129 \, \text{J/(g·°C)} \][/tex]
Now, we compare this result to the given choices:
[tex]\[ 0.00775 \, \text{J/(g·°C)} \][/tex]
[tex]\[ 0.0600 \, \text{J/(g·°C)} \][/tex]
[tex]\[ 0.129 \, \text{J/(g·°C)} \][/tex]
[tex]\[ 0.155 \, \text{J/(g·°C)} \][/tex]
The specific heat capacity of the substance is approximately [tex]\( \boxed{0.129 \, \text{J/(g·°C)} } \)[/tex].
