To determine the specific heat capacity [tex]\(C_p\)[/tex] of the substance, we will use the formula:
[tex]\[ q = m C_p \Delta T \][/tex]
Here's a detailed breakdown of the solution:
1. Identify the known variables:
- Mass ([tex]\(m\)[/tex]): 0.158 kg
- Initial temperature ([tex]\(T_{initial}\)[/tex]): [tex]\(32.0 ^{\circ}C\)[/tex]
- Final temperature ([tex]\(T_{final}\)[/tex]): [tex]\(61.0 ^{\circ}C\)[/tex]
- Specific heat capacity options are given in [tex]\(J/(g \cdot ^{\circ}C)\)[/tex].
2. Convert mass to grams:
Since the specific heat capacity values are in [tex]\(J/(g \cdot ^{\circ}C)\)[/tex], we need to convert the mass from kg to grams:
[tex]\[ 0.158 \, \text{kg} = 158 \, \text{g} \][/tex]
3. Calculate the temperature change ([tex]\(\Delta T\)[/tex]):
[tex]\[ \Delta T = T_{final} - T_{initial} \][/tex]
[tex]\[ \Delta T = 61.0 ^{\circ}C - 32.0 ^{\circ}C = 29.0 ^{\circ}C \][/tex]
4. Determine the heat added ([tex]\(q\)[/tex]):
Using the specific heat capacity value [tex]\(0.171 \, J/(g \cdot ^{\circ}C)\)[/tex]:
[tex]\[ q = m \cdot C_p \cdot \Delta T \][/tex]
[tex]\[ q = 158 \, \text{g} \cdot 0.171 \, J/(g \cdot ^{\circ}C) \cdot 29.0 ^{\circ}C \][/tex]
[tex]\[ q \approx 783.522 \, J \][/tex]
5. Calculate the specific heat capacity ([tex]\(C_p\)[/tex]):
Rearrange the formula to solve for [tex]\(C_p\)[/tex]:
[tex]\[ C_p = \frac{q}{m \cdot \Delta T} \][/tex]
Plugging in the known values:
[tex]\[ C_p = \frac{783.522 \, J}{158 \, \text{g} \cdot 29.0 ^{\circ}C} \][/tex]
[tex]\[ C_p \approx 0.171 \, J/(g \cdot ^{\circ}C) \][/tex]
Hence, the specific heat capacity of the substance is [tex]\(0.171 \, J/(g \cdot ^{\circ}C)\)[/tex]. This answer matches one of the given options, affirming our calculations and establishing that the correct answer is:
[tex]\[ \boxed{0.171 \, J/(g \cdot ^{\circ}C)} \][/tex]
