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Chapter 5: Problem 49

(a) Derive an equation to convert the specific heat of a pure substance to its molar heat capacity. (b) The specific heat of aluminum is $0.9 \mathrm{~J} /(\mathrm{g} \cdot \mathrm{K}) .$ Calculate its molar heat capacity. (c) If you know the specific heat of aluminum, what additional information do you need to calculate the heat capacity of a particular piece of an aluminum component?

Short Answer

Expert verified
(a) The equation to convert specific heat (Cₚ) of a pure substance to its molar heat capacity (Cₚ,m) is: Cₚ,m = Cₚ × M, where M is the molar mass of the substance. (b) The molar heat capacity of aluminum is approximately 24.3 J/(mol·K), calculated using its specific heat (0.9 J/(g·K)) and molar mass (26.98 g/mol). (c) To calculate the heat capacity of a particular piece of an aluminum component, we need the mass of the aluminum piece in addition to the specific heat of aluminum.

Step by step solution

01

(a) Deriving the conversion equation

To convert specific heat (Cₚ; in J/(g·K) ) of a pure substance to its molar heat capacity (Cₚ,m; in J/(mol·K)), we can use the relationship: Cₚ,m = Cₚ × M where M is the molar mass of the substance (in g/mol).
02

(b) Calculating the molar heat capacity of aluminum

First, we need to know the molar mass of aluminum. From the periodic table, the molar mass of aluminum is 26.98 g/mol. Given the specific heat of aluminum as 0.9 J/(g·K), we can calculate the molar heat capacity of aluminum using the conversion equation derived in step (a): Cₚ,m = Cₚ × M Cₚ,m = \( 0.9 \frac{J}{g \cdot K} \times 26.98 \frac{g}{mol} \) Cₚ,m = \( 24.282 \frac{J}{mol \cdot K} \) Hence, the molar heat capacity of aluminum is approximately 24.3 J/(mol·K).
03

(c) Additional information required

To calculate the heat capacity of a particular piece of an aluminum component with a known specific heat, we also need to know the mass of the piece. The heat capacity can then be calculated as follows: Heat Capacity = mass × specific heat where mass is the mass of the aluminum piece (in grams) and specific heat is the specific heat of aluminum (in J/(g·K)).

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Most popular questions from this chapter

A sample of a hydrocarbon is combusted completely in \(\mathrm{O}_{2}(g)\) to produce $21.83 \mathrm{~g} \mathrm{CO}_{2}(g), 4.47 \mathrm{~g} \mathrm{H}_{2} \mathrm{O}(g),\( and \)311 \mathrm{~kJ}$ of heat. (a) What is the mass of the hydrocarbon sample that was combusted? (b) What is the empirical formula of the hydrocarbon? (c) Calculate the value of \(\Delta H_{f}^{\circ}\) per empiricalformula unit of the hydrocarbon. (d) Do you think that the hydrocarbon is one of those listed in Appendix C? Explain your answer.

Atomic hydrogen (H) is used in welding (AHW). The atoms recombine to hydrogen molecules with a large release of heat according to the following reaction: $$ 2 \mathrm{H}(g) \longrightarrow \mathrm{H}_{2}(g) $$ (a) Using the thermodynamic data in Appendix C, calculate the enthalpy change for this reaction per mole of \(\mathrm{H}_{2}\). (b) Which has the higher enthalpy under these conditions, \(2 \mathrm{H}(g)\) or \(\mathrm{H}_{2}(g) ?\)

Calcium carbide \(\left(\mathrm{CaC}_{2}\right)\) reacts with water to form acetylene \(\left(\mathrm{C}_{2} \mathrm{H}_{2}\right)\) and \(\mathrm{Ca}(\mathrm{OH})_{2}\). From the following enthalpy of reaction data and data in Appendix C, calculate \(\Delta H_{f}^{\circ}\) for \(\mathrm{CaC}_{2}(s);\) $$ \begin{aligned} \mathrm{CaC}_{2}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{Ca}(\mathrm{OH})_{2}(s)+\mathrm{C}_{2} \mathrm{H}_{2}(g) & \\ \Delta H^{\circ}=&-127.2 \mathrm{~kJ} \end{aligned} $$

When an 18.6-g sample of solid potassium hydroxide dissolves in $200.0 \mathrm{~g}$ of water in a coffee-cup calorimeter (Figure 5.18), the temperature rises from 23.7 to \(44.5^{\circ} \mathrm{C}\). (a) Calculate the quantity of heat (in kJ) released in the reaction. (b) Using your result from part (a), calculate \(\Delta H\) (in kJ/mol KOH) for the solution process. Assume that the specific heat of the solution is the same as that of pure water.

During a deep breath, our lungs expand about \(2.0 \mathrm{~L}\) against an external pressure of \(101.3 \mathrm{kPa}\). How much work is involved in this process (in J)?
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