A sample of tin [tex]\((Cp = 0.227 \, J/g \cdot {}^{\circ}C)\)[/tex] is placed in a freezer. Its temperature decreases from [tex]\(15.0^{\circ}C\)[/tex] to [tex]\(-10.0^{\circ}C\)[/tex] as it releases [tex]\(543 \, J\)[/tex] of energy. What is the mass of the sample? Round your answer to three significant figures.

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

[tex]\[ \boxed{\text{g}} \][/tex]



Answer :

To determine the mass of the tin sample, we start by clearly understanding and applying the problem's given details alongside the formula [tex]\( q = m C_p \Delta T \)[/tex].

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

Step-by-step solution:

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

[tex]\[ \Delta T = T_{\text{final}} - T_{\text{initial}} \][/tex]

[tex]\[ \Delta T = -10.0^{\circ}C - 15.0^{\circ}C \][/tex]

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

2. Rearrange the formula [tex]\( q = m C_p \Delta T \)[/tex] to solve for mass ([tex]\( m \)[/tex]):

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

3. Substitute the known values into the formula:

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

4. Perform the calculation:

[tex]\[ m = \frac{543}{0.227 \times (-25.0)} \][/tex]

[tex]\[ m = \frac{543}{-5.675} \][/tex]

[tex]\[ m \approx -95.68281938325991 \, \text{g} \][/tex]

5. Round the mass to three significant figures:

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

However, since mass cannot be negative, it indicates we consider the absolute value when dealing with magnitude:

[tex]\[ m = 95.683 \, \text{g} \][/tex]

Therefore, the mass of the tin sample is [tex]\(\boxed{95.683} \, \text{g}\)[/tex].