A sample of tin (Cp = 0.227 J/g·°C) is placed in a freezer. Its temperature decreases from 15.0°C to -10.0°C, releasing 543 J of energy.

What is the mass of the sample? Round your answer to three significant figures.

Use the formula [tex]\( q = mc\Delta T \)[/tex].



Answer :

Certainly! Let's solve the problem step by step.

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Given Data:
- Specific heat capacity of tin, [tex]\( Cp = 0.227 \, \text{J/g} \cdot ^\circ\text{C} \)[/tex]
- Initial temperature, [tex]\( T_{\text{initial}} = 15.0 \, ^\circ\text{C} \)[/tex]
- Final temperature, [tex]\( T_{\text{final}} = -10.0 \, ^\circ\text{C} \)[/tex]
- Energy released, [tex]\( q = 543 \, \text{J} \)[/tex]

Formula to Use:
[tex]\[ q = m \cdot Cp \cdot \Delta T \][/tex]
where:
- [tex]\( q \)[/tex] is the energy released or absorbed (in Joules)
- [tex]\( m \)[/tex] is the mass (in grams)
- [tex]\( Cp \)[/tex] is the specific heat capacity (in J/g·°C)
- [tex]\( \Delta T \)[/tex] is the change in temperature (in °C)

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Step-by-Step Solution:

1. Calculate the change in temperature [tex]\( \Delta T \)[/tex]:

[tex]\[ \Delta T = T_{\text{final}} - T_{\text{initial}} = -10.0 \, ^\circ\text{C} - 15.0 \, ^\circ\text{C} = -25.0 \, ^\circ\text{C} \][/tex]

The change in temperature is [tex]\( -25.0 \, ^\circ\text{C} \)[/tex].

2. Rearrange the formula to solve for mass [tex]\( m \)[/tex]:

[tex]\[ m = \frac{q}{Cp \cdot \Delta T} \][/tex]

3. Plug in the known values:

- [tex]\( q = 543 \, \text{J} \)[/tex]
- [tex]\( Cp = 0.227 \, \text{J/g} \cdot ^\circ\text{C} \)[/tex]
- [tex]\( \Delta T = -25.0 \, ^\circ\text{C} \)[/tex]

Note that since [tex]\(\Delta T\)[/tex] is negative, and we are interested in an absolute mass value, we'll take the absolute value of [tex]\(\Delta T\)[/tex] in the calculation.

[tex]\[ m = \frac{543 \, \text{J}}{0.227 \, \text{J/g} \cdot ^\circ\text{C} \times |-25.0 \, ^\circ\text{C}|} = \frac{543}{0.227 \times 25.0} \][/tex]

4. Calculate the mass:

[tex]\[ m = \frac{543}{5.675} \approx 95.683 \, \text{g} \][/tex]

5. Round the mass to three significant figures:

[tex]\[ m \approx 95.683 \, \text{g} \rightarrow 95.7 \, \text{g} \][/tex]

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Final Answer:

The mass of the tin sample is [tex]\( 95.7 \)[/tex] grams (rounded to three significant figures).