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A sample of an unknown substance has a mass of 0.158 kg. If 2,510.0 J of heat is required to heat the substance from 32.0°C to 61.0°C, what is the specific heat of the substance?

Use [tex]\( q = m C_p \Delta T \)[/tex].

A. [tex]\( 0.171 \, \text{J} / (\text{g} \cdot {}^\circ \text{C}) \)[/tex]
B. [tex]\( 0.548 \, \text{J} / (\text{g} \cdot {}^\circ \text{C}) \)[/tex]
C. [tex]\( 15.9 \, \text{J} / (\text{g} \cdot {}^\circ \text{C}) \)[/tex]
D. [tex]\( 86.6 \, \text{J} / (\text{g} \cdot {}^\circ \text{C}) \)[/tex]


Sagot :

Sure, let's solve this problem step by step.

We are given the following information:
- Mass of the substance, [tex]\( m = 0.158 \, \text{kg} \)[/tex]
- Heat added, [tex]\( q = 2510.0 \, \text{J} \)[/tex]
- Initial temperature, [tex]\( T_{\text{initial}} = 32.0 \, ^\circ \text{C} \)[/tex]
- Final temperature, [tex]\( T_{\text{final}} = 61.0 \, ^\circ \text{C} \)[/tex]

We need to find the specific heat capacity [tex]\( C_p \)[/tex] of the substance.

First, calculate the change in temperature ([tex]\( \Delta T \)[/tex]):
[tex]\[ \Delta T = T_{\text{final}} - T_{\text{initial}} \][/tex]
[tex]\[ \Delta T = 61.0^\circ \text{C} - 32.0^\circ \text{C} \][/tex]
[tex]\[ \Delta T = 29.0^\circ \text{C} \][/tex]

Next, we need to convert the mass from kilograms to grams since the potential answers for specific heat capacity are given in units of [tex]\( J / (g \cdot ^\circ C) \)[/tex]:
[tex]\[ 1 \, \text{kg} = 1000 \, \text{g} \][/tex]
[tex]\[ m = 0.158 \, \text{kg} \times 1000 \, \text{g/kg} = 158.0 \, \text{g} \][/tex]

Now, use the formula [tex]\( q = m C_p \Delta T \)[/tex] to solve for [tex]\( C_p \)[/tex]:
[tex]\[ C_p = \frac{q}{m \Delta T} \][/tex]
[tex]\[ C_p = \frac{2510.0 \, \text{J}}{158.0 \, \text{g} \times 29.0^\circ \text{C}} \][/tex]

Perform the division:
[tex]\[ C_p = \frac{2510.0}{4582.0} \][/tex]
[tex]\[ C_p \approx 0.5478 \, J / (g \cdot ^\circ C) \][/tex]

Thus, the specific heat capacity of the substance is approximately [tex]\( 0.548 \, J / (g \cdot ^\circ C) \)[/tex]. The closest answer is:
[tex]\[ \boxed{0.548 \, J / (g \cdot ^\circ C)} \][/tex]
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