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Chapter 7: Problem 54

A biology experiment requires the preparation of a water bath at \(37.0^{\circ} \mathrm{C}\) (body temperature). The temperature of the cold tap water is \(22.0^{\circ} \mathrm{C},\) and the temperature of the hot tap water is \(55.0^{\circ} \mathrm{C} .\) If a student starts with \(90.0 \mathrm{g}\) cold water, what mass of hot water must be added to reach \(37.0^{\circ} \mathrm{C} ?\)

Short Answer

Expert verified
The student needs to add \(75.0 grams\) of hot water to the \(90.0 grams\) of cold water to achieve a final temperature of \(37.0^{\circ} C\).

Step by step solution

01

Write down the specific heat formula and conservation of energy principle

The heat q gained or lost by a substance can be calculated using the equation: \(q = mcΔT\), where m is the mass of the substance, c is the specific heat capacity, and ΔT is the change in temperature. According to the conservation of energy principle, the heat gained by the cold water must be equal to the heat lost by the hot water. Thus, we have: \(m_{cold}c_{water}(T_{final} - T_{cold}) = m_{hot}c_{water}(T_{hot} - T_{final})\)
02

Simplify the formula

Since we're working with water, the specific heat capacity, \(c_{water}\), is the same for both cold and hot water. Therefore, we can simplify the equation by dividing both sides by \(c_{water}\): \(m_{cold}(T_{final} - T_{cold}) = m_{hot}(T_{hot} - T_{final})\)
03

Substitute the given values

Now, we can substitute the given values in the formula: • \(m_{cold} = 90.0 g\) • \(T_{cold} = 22.0^{\circ} C\) • \(T_{hot} = 55.0^{\circ} C\) • \(T_{final} = 37.0^{\circ} C\) \(90.0(37.0 - 22.0) = m_{hot}(55.0 - 37.0)\)
04

Solve for m_hot

Solve the equation for \(m_{hot}\): \(90.0(15.0) = m_{hot}(18.0)\) \(1350 = 18m_{hot}\) Now, to find the mass of hot water, divide both sides by 18: \(m_{hot} = \frac{1350}{18}\) \(m_{hot} = 75.0 g\)
05

State the final answer

The student needs to add 75.0 grams of hot water to the 90.0 grams of cold water to achieve a final temperature of 37.0°C.

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Most popular questions from this chapter

In a bomb calorimeter, the reaction vessel is surrounded by water that must be added for each experiment. since the amount of water is not constant from experiment to experiment, the mass of water must be measured in each case. The heat capacity of the calorimeter is broken down into two parts: the water and the calorimeter components. If a calorimeter contains \(1.00 \mathrm{kg}\) water and has a total heat capacity of \(10.84 \mathrm{kJ} /^{\circ} \mathrm{C},\) what is the heat capacity of the calorimeter components?

The bomb calorimeter in Exercise 102 is filled with 987 g water. The initial temperature of the calorimeter contents is \(23.32^{\circ} \mathrm{C} . \mathrm{A}\) \(1.056-\mathrm{g}\) sample of benzoic acid \(\left(\Delta E_{\text {comb }}=-26.42 \mathrm{kJ} / \mathrm{g}\right)\) is combusted in the calorimeter. What is the final temperature of the calorimeter contents?

You have a 1.00 -mole sample of water at \(-30 .^{\circ} \mathrm{C}\) and you heat it until you have gaseous water at \(140 .^{\circ} \mathrm{C}\). Calculate \(q\) for the entire process. Use the following data. Specific heat capacity of ice \(=2.03 \mathrm{J} /^{\circ} \mathrm{C} \cdot \mathrm{g}\) Specific heat capacity of water \(=4.18 \mathrm{J} /^{\circ} \mathrm{C} \cdot \mathrm{g}\) Specific heat capacity of steam \(=2.02 \mathrm{J} /^{\circ} \mathrm{C} \cdot \mathrm{g}\) $$\begin{array}{ll}\mathrm{H}_{2} \mathrm{O}(s) \longrightarrow \mathrm{H}_{2} \mathrm{O}(l) & \Delta H_{\text {fusion }}=6.02 \mathrm{kJ} / \mathrm{mol}\left(\mathrm{at} 0^{\circ} \mathrm{C}\right) \\\\\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{H}_{2} \mathrm{O}(g) & \Delta H_{\text {vaporization }}=40.7 \mathrm{kJ} / \mathrm{mol}\left(\text { at } 100 .^{\circ} \mathrm{C}\right)\end{array}$$

Consider the following statements: "Heat is a form of energy, and energy is conserved. The heat lost by a system must be equal to the amount of heat gained by the surroundings. Therefore, heat is conserved." Indicate everything you think is correct in these statements. Indicate everything you think is incorrect. Correct the incorrect statements and explain.

Are the following processes exothermic or endothermic? a. the combustion of gasoline in a car engine b. water condensing on a cold pipe c. \(\mathrm{CO}_{2}(s) \longrightarrow \mathrm{CO}_{2}(g)\) d. \(\mathrm{F}_{2}(g) \longrightarrow 2 \mathrm{F}(g)\)
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