Calculate the specific heat (\rm{J/g \; ^\circ C}) for a 18.5-\rm g sample of tin that absorbs 183 {\rm J} when temperature increases from 35.0 ^\circ \rm C to 78.6 ^\circ \rm C.

To calculate the specific heat (\rm{J/g \; ^\circ C}) for a sample of tin, first, we need to understand the formula for specific heat.

The formula for specific heat is:

\[Q = mc\Delta T\]

Where:
- \(Q\) is the heat energy absorbed (in joules)
- \(m\) is the mass of the sample (in grams)
- \(c\) is the specific heat capacity (in \rm{J/g \; ^\circ C})
- \(\Delta T\) is the change in temperature (in degrees Celsius)

In this case, we already have the values for \(Q\), \(m\), and the initial and final temperatures. We need to solve for \(c\).

Given values:
\(m = 18.5 \, \rm g\)
\(Q = 183 \, \rm J\)
Initial temperature, \(T_1 = 35.0 \, ^\circ \rm C\)
Final temperature, \(T_2 = 78.6 \, ^\circ \rm C\)

So we can rearrange the formula to solve for specific heat \(c\):

\[c = \frac{Q}{m \cdot \Delta T}\]

Now, substitute the given values into the formula:

\[c = \frac{183 \, \rm J}{18.5 \, \rm g \cdot (78.6 \, ^\circ \rm C - 35.0 \, ^\circ \rm C)}\]

Let's calculate it: