To determine the molar heat capacity of $\text{C}_4\text{H}_8\text{O}_2$, we'll use the heat transfer formula:

$q = mc\Delta T$

where:

• $q$ is the amount of heat added (2.27 × 10³ J),
• $m$ is the mass of the sample (129.0 g),
• $c$ is the specific heat capacity (J/g·°C),
• $\Delta T$ is the change in temperature (final temperature - initial temperature).

Step 1: Calculate $\Delta T$

$\Delta T = 49.8^\circ \text{C} - 38.5^\circ \text{C} = 11.3^\circ \text{C}$

Step 2: Solve for Specific Heat Capacity $c$

Rearrange the formula to solve for $c$:

$c = \frac{q}{m \Delta T}$

Substitute the values:

c = \frac{2.27 \times 10^3 \, \text{J}}{129.0 \, \text{g} \times 11.3 \, ^\circ\text{C}}

Calculating this gives:

$c \approx 1.56 \, \text{J/g} \cdot ^\circ\text{C}$

Step 3: Convert to Molar Heat Capacity

To find the molar heat capacity, we need the molar mass of $\text{C}_4\text{H}_8\text{O}_2$.

1. Carbon (C): $4 \times 12.01 = 48.04$
2. Hydrogen (H): $8 \times 1.008 = 8.064$
3. Oxygen (O): $2 \times 16.00 = 32.00$

$\text{Molar mass of } \text{C}_4\text{H}_8\text{O}_2 = 48.04 + 8.064 + 32.00 = 88.104 \, \text{g/mol}$

Now we calculate the molar heat capacity ($C_m$) by multiplying the specific heat capacity by the molar mass:

$C_m = c \times \text{Molar mass}$

$C_m = 1.56 \, \text{J/g} \cdot ^\circ\text{C} \times 88.104 \, \text{g/mol}$

Calculating this:

$C_m \approx 137 \, \text{J/mol} \cdot ^\circ\text{C}$

Answer

The chemist can report the molar heat capacity of $\text{C}_4\text{H}_8\text{O}_2$ as:

$C_m = 137 \, \text{J/mol} \cdot ^\circ\text{C}$

This answer has the correct number of significant figures (three), based on the precision of the given data.

Would you like additional details on any of these steps?

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1. What is the difference between specific and molar heat capacity?
2. How is significant figure determination handled in scientific calculations?
3. How does temperature change affect heat capacity calculations?
4. What would be the effect of using a different sample mass?
5. How does molecular composition affect the molar heat capacity?

Tip: Always ensure units match up in each calculation step for accurate results.