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Chapter 6: Problem 29

Calculate \(\Delta E\) for each of the following. a. \(q=-47 \mathrm{~kJ}, w=+88 \mathrm{~kJ}\) b. \(q=+82 \mathrm{~kJ}, w=-47 \mathrm{~kJ}\) c. \(q=+47 \mathrm{~kJ}, w=0\) d. In which of these cases do the surroundings do work on the system?

Short Answer

Expert verified
The change in internal energy (\(\Delta E\)) for each case is calculated as follows: a. \(\Delta E = 41 \mathrm{~kJ}\) b. \(\Delta E = 35 \mathrm{~kJ}\) c. \(\Delta E = 47 \mathrm{~kJ}\) The surroundings do work on the system only in case a.

Step by step solution

01

Case a: Calculate \(\Delta E\)

To calculate the change in internal energy for case a, we will plug in the given values of \(q\) and \(w\) into the formula: \[\Delta E = (-47 \mathrm{~kJ}) + (+88 \mathrm{~kJ})\] Now, we simply add the two values together: \[\Delta E = 41 \mathrm{~kJ}\]
02

Case b: Calculate \(\Delta E\)

For case b, plug in the given values of \(q\) and \(w\) into the formula: \[\Delta E = (+82 \mathrm{~kJ}) + (-47 \mathrm{~kJ})\] Now, add the two values together: \[\Delta E = 35 \mathrm{~kJ}\]
03

Case c: Calculate \(\Delta E\)

For case c, plug in the given values of \(q\) and \(w\) into the formula: \[\Delta E = (+47 \mathrm{~kJ}) + (0)\] Since work done (\(w\)) is zero, the change in internal energy is equal to the heat added to the system: \[\Delta E = 47 \mathrm{~kJ}\]
04

Identify cases where surroundings do work on the system

The surroundings do work on the system when the work (\(w\)) is positive. Looking at the given values for each case, we find that: - In case a, \(w = +88\) kJ (positive) - In case b, \(w = -47\) kJ (negative) - In case c, \(w = 0\) kJ (zero) Thus, the surroundings do work on the system only in case a.

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Most popular questions from this chapter

A sample consisting of \(22.7 \mathrm{~g}\) of a nongaseous, unstable compound \(\mathrm{X}\) is placed inside a metal cylinder with a radius of \(8.00 \mathrm{~cm}\), and a piston is carefully placed on the surface of the compound so that, for all practical purposes, the distance between the bottom of the cylinder and the piston is zero. (A hole in the piston allows trapped air to escape as the piston is placed on the compound; then this hole is plugged so that nothing inside the cylinder can escape.) The piston-and-cylinder apparatus is carefully placed in \(10.00 \mathrm{~kg}\) water at \(25.00^{\circ} \mathrm{C}\). The barometric pressure is 778 torr. When the compound spontaneously decomposes, the piston moves up, the temperature of the water reaches a maximum of \(29.52^{\circ} \mathrm{C}\), and then it gradually decreases as the water loses heat to the surrounding air. The distance between the piston and the bottom of the cylinder, at the maximum temperature, is \(59.8 \mathrm{~cm}\). Chemical analysis shows that the cylinder contains \(0.300 \mathrm{~mol}\) carbon dioxide, \(0.250\) mol liquid water, \(0.025\) mol oxygen gas, and an undetermined amount of a gaseous element \(\mathrm{A}\). It is known that the enthalpy change for the decomposition of \(X\), according to the reaction described above, is \(-1893\) \(\mathrm{kJ} / \mathrm{mol} \mathrm{X}\). The standard enthalpies of formation for gaseous carbon dioxide and liquid water are \(-393.5 \mathrm{~kJ} / \mathrm{mol}\) and \(-286 \mathrm{~kJ} / \mathrm{mol}\), respectively. The heat capacity for water is \(4.184 \mathrm{~J} /{ }^{\circ} \mathrm{C} \cdot \mathrm{g}\). The conversion factor between \(\mathrm{L} \cdot \mathrm{atm}\) and \(\mathrm{J}\) can be determined from the two values for the gas constant \(R\), namely, \(0.08206 \mathrm{~L}\). \(\mathrm{atm} / \mathrm{K} \cdot \mathrm{mol}\) and \(8.3145 \mathrm{~J} / \mathrm{K} \cdot \mathrm{mol}\). The vapor pressure of water at \(29.5^{\circ} \mathrm{C}\) is 31 torr. Assume that the heat capacity of the pistonand-cylinder apparatus is negligible and that the piston has negligible mass. Given the preceding information, determine a. The formula for \(\mathrm{X}\). b. The pressure-volume work (in \(\mathrm{kJ}\) ) for the decomposition of the \(22.7-\mathrm{g}\) sample of \(\mathrm{X}\). c. The molar change in internal energy for the decomposition of \(X\) and the approximate standard enthalpy of formation for \(X\).

The equation for the fermentation of glucose to alcohol and carbon dioxide is $$ \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(a q) \longrightarrow 2 \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(a q)+2 \mathrm{CO}_{2}(g) $$ The enthalpy change for the reaction is \(-67 \mathrm{~kJ}\). Is the reaction exothermic or endothermic? Is energy, in the form of heat, absorbed or evolved as the reaction occurs?

If the internal energy of a thermodynamic system is increased by \(300 .\) J while \(75 \mathrm{~J}\) of expansion work is done, how much heat was transferred and in which direction, to or from the system?

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 preparation of \(\mathrm{NO}_{2}(g)\) from \(\mathrm{N}_{2}(g)\) and \(\mathrm{O}_{2}(g)\) is an endothermic reaction: $$ \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{NO}_{2}(g) \text { (unbalanced) } $$ The enthalpy change of reaction for the balanced equation (with lowest whole- number coefficients) is \(\Delta H=67.7 \mathrm{~kJ}\). If \(2.50 \times\) \(10^{2} \mathrm{~mL} \mathrm{~N}_{2}(g)\) at \(100 .{ }^{\circ} \mathrm{C}\) and \(3.50\) atm and \(4.50 \times 10^{2} \mathrm{~mL} \mathrm{O}_{2}(g)\) at \(100 .{ }^{\circ} \mathrm{C}\) and \(3.50 \mathrm{~atm}\) are mixed, what amount of heat is necessary to synthesize the maximum yield of \(\mathrm{NO}_{2}(g)\) ?
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