Questions: A chemical reaction takes place inside a flask submerged in a water bath. The water bath contains 7.80 kg of water at 27.4 °C. During the reaction 57.5 kJ of heat flows out of the flask and into the bath. Calculate the new temperature of the water bath. You can assume the specific heat capacity of water under these conditions is 4.18 J ⋅ g^-1 ⋅ K^-1. Round your answer to 3 significant digits.

A chemical reaction takes place inside a flask submerged in a water bath. The water bath contains 7.80 kg of water at 27.4 °C. During the reaction 57.5 kJ of heat flows out of the flask and into the bath.

Calculate the new temperature of the water bath. You can assume the specific heat capacity of water under these conditions is 4.18 J ⋅ g^-1 ⋅ K^-1. Round your answer to 3 significant digits.

Transcript text: A chemical reaction takes place inside a flask submerged in a water bath. The water bath contains 7.80 kg of water at $27.4{ }^{\circ} \mathrm{C}$. During the reaction 57.5 kJ of heat flows out of the flask and into the bath. Calculate the new temperature of the water bath. You can assume the specific heat capacity of water under these conditions is $4.18 \mathrm{~J} \cdot \mathrm{~g}^{-1} \cdot \mathrm{~K}^{-1}$. Round your answer to 3 significant digits.

Solution

Solution Steps

Step 1: Convert Mass of Water to Grams

First, we need to convert the mass of the water from kilograms to grams because the specific heat capacity is given in $\mathrm{J} \cdot \mathrm{g}^{-1} \cdot \mathrm{K}^{-1}$.

\[ 7.80 \, \text{kg} = 7800 \, \text{g} \]

Step 2: Calculate the Heat Added to the Water

The heat added to the water bath is given as 57.5 kJ. We need to convert this to joules.

\[ 57.5 \, \text{kJ} = 57500 \, \text{J} \]

Step 3: Use the Specific Heat Formula

We use the formula for heat transfer:

\[ q = m \cdot c \cdot \Delta T \]

where:

$ q $ is the heat added (57500 J),
$ m $ is the mass of the water (7800 g),
$ c $ is the specific heat capacity of water ($4.18 \, \mathrm{J} \cdot \mathrm{g}^{-1} \cdot \mathrm{K}^{-1}$),
$ \Delta T $ is the change in temperature.

Rearranging the formula to solve for $\Delta T$:

\[ \Delta T = \frac{q}{m \cdot c} = \frac{57500 \, \text{J}}{7800 \, \text{g} \cdot 4.18 \, \mathrm{J} \cdot \mathrm{g}^{-1} \cdot \mathrm{K}^{-1}} \]

Step 4: Calculate the Change in Temperature

Perform the calculation:

\[ \Delta T = \frac{57500}{32604} \approx 1.763 \, \text{K} \]

Step 5: Determine the New Temperature

The initial temperature of the water bath is $27.4 \, ^\circ \text{C}$. Adding the change in temperature:

\[ T_{\text{new}} = 27.4 \, ^\circ \text{C} + 1.763 \, \text{K} \approx 29.2 \, ^\circ \text{C} \]