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Chapter 23: Problem 23

A certain substance has a molar mass of \(51.4 \mathrm{~g} / \mathrm{mol}\). When \(320 \mathrm{~J}\) of heat are added to a 37.1-g sample of this material, its temperature rises from \(26.1\) to \(42.0^{\circ} \mathrm{C}\). \((a)\) Find the specific heat of the substance. (b) How many moles of the substance are present? (c) Calculate the molar heat capacity of the substance.

Short Answer

Expert verified
The specific heat of the substance (c) can be calculated using the values given. Use the calculated c to find the number of moles (n) of the substance. Lastly, use the derived n and the provided values to find the molar heat capacity (C).

Step by step solution

01

Compute for the Specific Heat

The specific heat of a substance can be calculated using the formula \(c = Q / (m \cdot \Delta T)\), where \(Q\) is the heat energy added, \(m\) is the mass of the substance, and \(\Delta T\) is the change in temperature. Substituting the given values: \(Q = 320 \mathrm{~J}\), \(m = 37.1 \mathrm{~g}\), and \(\Delta T = 42.0^{\circ} \mathrm{C}-26.1^{\circ} \mathrm{C}\) into the formula will yield the specific heat.
02

Calculate the Number of Moles

The number of moles of a substance can be calculated using the formula \(n = m / M\), where \(m\) is the mass of the substance and \(M\) is the molar mass. The substance's mass is given as 37.1 g and the molar mass is given as 51.4 g/mol. Substituting these values into the formula will yield the number of moles.
03

Determine the Molar Heat Capacity

The molar heat capacity can be calculated using the formula \(C = Q / (n \cdot \Delta T)\), where \(Q\) is the heat energy added, \(n\) is the number of moles, and \(\Delta T\) is the change in temperature. Here, \(Q = 320 \mathrm{~J}\), \(n\) is the moles calculated in step 2, and \(\Delta T = 42.0^{\circ} \mathrm{C}-26.1^{\circ} \mathrm{C}\). Substituting these values into the formula will yield the molar heat capacity.

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Most popular questions from this chapter

Ice has formed on a shallow pond and a steady state has been reached with the air above the ice at \(-5.20^{\circ} \mathrm{C}\) and the bottom of the pond at \(3.98^{\circ} \mathrm{C}\). If the total depth of ice \(+\) water is \(1.42 \mathrm{~m}\), how thick is the ice? (Assume that the thermal conductivities of ice and water are \(1.67\) and \(0.502 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), respectively.)

An ideal gas experiences an adiabatic compression from \(p=122 \mathrm{kPa}, V=10.7 \mathrm{~m}^{3}, T=-23.0^{\circ} \mathrm{C}\) to \(p=1450 \mathrm{kPa}\) \(V=1.36 \mathrm{~m}^{3} .(a)\) Calculate the value of \(\gamma .(b)\) Find the final temperature. ( \(c\) ) How many moles of gas are present? \((d)\) What is the total translational kinetic energy per mole before and after the compression? (e) Calculate the ratio of the \(\mathrm{rms}\) speed before to that after the compression.

A container holds a mixture of three nonreacting gases: \(n_{1}\) moles of the first gas with molar specific heat at constant volume \(C_{1}\), and so on. Find the molar specific heat at constant volume of the mixture, in terms of the molar specific heats and quantities of the three separate gases.

a) Calculate the rate at which body heat flows out through the clothing of a skier, given the following data: the body surface area is \(1.8 \mathrm{~m}^{2}\) and the clothing is \(1.2 \mathrm{~cm}\) thick; skin surface temperature is \(33^{\circ} \mathrm{C}\), whereas the outer surface of the clothing is at \(1.0^{\circ} \mathrm{C} ;\) the thermal conductivity of the clothing is \(0.040 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K} .(b)\) How would the answer change if, after a fall, the skier's clothes become soaked with water? Assume that the thermal conductivity of water is \(0.60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\)

A sample of \(n\) moles of an ideal gas undergoes an isothermal expansion. Find the heat flow into the gas in terms of the initial and final volumes and the temperature.
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