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Chapter 15: Problem 69

Estimate the entropy change of \(850 \mathrm{g}\) of water when it is heated from \(20.0^{\circ} \mathrm{C}\) to \(50.0^{\circ} \mathrm{C} .\) [Hint: Assume that the heat flows into the water at an average temperature. \(]\)

Short Answer

Expert verified
The estimated entropy change is approximately 349 J/K when 850g of water is heated from 20.0°C to 50.0°C.

Step by step solution

01

Convert temperatures to Kelvin

First, we need to convert the given temperatures from Celsius to Kelvin. To do this, we will add 273.15 to each Celsius temperature: \(T_1 = 20.0 + 273.15 = 293.15 \mathrm{K}\) \(T_2 = 50.0 + 273.15 = 323.15 \mathrm{K}\)
02

Calculate the average temperature

We will now calculate the average temperature during the heating process: \(T_\text{avg} = \frac{T_1 + T_2}{2} = \frac{293.15 + 323.15}{2} = 308.15 \mathrm{K}\)
03

Calculate the entropy change

We can now calculate the entropy change using the formula: \(\Delta S = mC\log \frac{T_2}{T_1} = (850 \mathrm{g})(4.18 \mathrm{J / g \cdot K})\log \frac{323.15 \mathrm{K}}{293.15 \mathrm{K}}\) Now, compute the logarithm and multiply the resulting value with the mass and specific heat capacity: \(\Delta S = (850 \mathrm{g})(4.18 \mathrm{J / g \cdot K})(0.0970) = 348.803 \mathrm{J/K}\)
04

Round the result

Finally, we will round the result to an appropriate number of significant figures (three): \(\Delta S \approx 349 \mathrm{J/K}\) The estimated entropy change of 850g of water when it is heated from \(20.0^{\circ} \mathrm{C}\) to \(50.0^{\circ} \mathrm{C}\) is approximately 349 J/K.

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Most popular questions from this chapter

An air conditioner whose coefficient of performance is 2.00 removes $1.73 \times 10^{8} \mathrm{J}$ of heat from a room per day. How much does it cost to run the air conditioning unit per day if electricity costs \(\$ 0.10\) per kilowatt-hour? (Note that 1 kilowatt-hour \(=3.6 \times 10^{6} \mathrm{J} .\) )
An ice cube at \(0.0^{\circ} \mathrm{C}\) is slowly melting. What is the change in the ice cube's entropy for each \(1.00 \mathrm{g}\) of ice that melts?
(a) How much heat does an engine with an efficiency of \(33.3 \%\) absorb in order to deliver \(1.00 \mathrm{kJ}\) of work? (b) How much heat is exhausted by the engine?
In a heat engine, 3.00 mol of a monatomic ideal gas, initially at 4.00 atm of pressure, undergoes an isothermal expansion, increasing its volume by a factor of 9.50 at a constant temperature of \(650.0 \mathrm{K} .\) The gas is then compressed at a constant pressure to its original volume. Finally, the pressure is increased at constant volume back to the original pressure. (a) Draw a \(P V\) diagram of this three-step heat engine. (b) For each step of this process, calculate the work done on the gas, the change in internal energy, and the heat transferred into the gas. (c) What is the efficiency of this engine?
List these in order of increasing entropy: (a) 1 mol of water at $20^{\circ} \mathrm{C}\( and 1 mol of ethanol at \)20^{\circ} \mathrm{C}$ in separate containers; (b) a mixture of 1 mol of water at \(20^{\circ} \mathrm{C}\) and 1 mol of ethanol at \(20^{\circ} \mathrm{C} ;\) (c) 0.5 mol of water at \(20^{\circ} \mathrm{C}\) and 0.5 mol of ethanol at \(20^{\circ} \mathrm{C}\) in separate containers; (d) a mixture of 1 mol of water at $30^{\circ} \mathrm{C}\( and 1 mol of ethanol at \)30^{\circ} \mathrm{C}$.
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