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Chapter 10: Problem 29

(a) What conditions are represented by the abbreviation STP? (b) What is the molar volume of an ideal gas at STP? (c) Room temperature is often assumed to be \(25^{\circ} \mathrm{C}\). Calculate the molar volume of an ideal gas at \(25^{\circ} \mathrm{C}\) and \(101.3 \mathrm{kPa}\) pressure. \((\mathbf{d})\) If you measure pressure in bars instead of atmospheres, calculate the corresponding value of \(R\) in \(\mathrm{L}\) -bar \(/ \mathrm{mol}-\mathrm{K}\).

Short Answer

Expert verified
(a) STP stands for Standard Temperature and Pressure, which represents a temperature of \(273.15 K\) and a pressure of \(1\) atm or \(101.3\) kPa. (b) The molar volume of an ideal gas at STP is \(22.4 \mathrm{L/mol}\). (c) The molar volume of an ideal gas at \(25^{\circ} \mathrm{C}\) and \(101.3\) kPa pressure is \(24.5 \mathrm{L/mol}\). (d) The value of R when measuring pressure in bars is \(0.0831 \mathrm{L \cdot bar / mol \cdot K}\).

Step by step solution

01

(a) Determine the conditions represented by STP

STP stands for Standard Temperature and Pressure. Under STP conditions, the temperature (T) is \(273.15 K\) and the pressure (P) is \(1\) atm or \(101.3\) kPa.
02

(b) Find the molar volume of an ideal gas at STP

We will use the ideal gas law, which is given by \(PV = nRT\), where P is pressure, V is volume, n is the number of moles, R is the universal gas constant, and T is the temperature. At STP, we have: - T = \(273.15 K\) - P = \(1\) atm = \(101.3\) kPa - For one mole of an ideal gas, n = \(1\) mol The value of R is \(0.0821 \mathrm{L} \cdot \mathrm{atm} / \mathrm{mol} \cdot K\) and we will find the volume V of the gas. So, the equation becomes: \(PV = RT\) Solving for V: \(V = \frac{RT}{P}\) Using the STP values: \(V = \frac{(0.0821 \mathrm{L} \cdot \mathrm{atm} / \mathrm{mol} \cdot K)(273.15 K)}{1 \mathrm{atm}}\) After the calculation, we get the molar volume of an ideal gas at STP as: \(V = 22.4 \mathrm{L/mol}\)
03

(c) Calculate the molar volume of an ideal gas at 25掳C and 101.3 kPa pressure

For this question, make sure to convert the temperature to Kelvin (K) by adding \(273.15\): \(25^{\circ} \mathrm{C} + 273.15 = 298.15 K\) Now we have: - T = \(298.15 K\) - P = \(101.3\) kPa To find the volume V, use the ideal gas law again: \(PV = nRT\) Since we are asked for molar volume, we use n = \(1\) mol. When pressure is given in kPa, we need to use a different value for the universal gas constant: \(R = 8.314 \mathrm{L} \cdot \mathrm{kPa} / \mathrm{mol} \cdot K\) Using the given temperature and pressure, the equation becomes: \(V = \frac{(8.314 \mathrm{L} \cdot \mathrm{kPa} / \mathrm{mol} \cdot K)(298.15 K)}{101.3 \mathrm{kPa}}\) After the calculation, we get the molar volume of an ideal gas at \(25^{\circ} \mathrm{C}\) and \(101.3\) kPa pressure as: \(V = 24.5 \mathrm{L/mol}\)
04

(d) Calculate the value of R in L-bar/mol-K if the pressure is measured in bars

We know that \(1 \mathrm{bar} = 100 \mathrm{kPa}\), so we need to convert the units of the gas constant R accordingly. Currently, we are given the value of R for the units kPa as \(R = 8.314 \mathrm{L} \cdot \mathrm{kPa} / \mathrm{mol} \cdot K\) To convert to L-bar/mol-K, we use the conversion factor: \(R' = \frac{8.314 \mathrm{L} \cdot \mathrm{kPa}}{1 \mathrm{mol} \cdot K} \times \frac{1 \mathrm{bar}}{100 \mathrm{kPa}}\) After the calculation, we get the value of R for pressure measured in bars as: \(R' = 0.0831 \mathrm{L \cdot bar / mol \cdot K}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Temperature and Pressure (STP)
When discussing gases in chemistry, we often refer to STP, which stands for Standard Temperature and Pressure. These conditions are commonly used as a reference to simplify calculations and express gas volumes.
At STP, the temperature is defined as exactly 273.15 Kelvin, equivalent to 0 degrees Celsius. This is the freezing point of water. The pressure at STP is standardized at 1 atmosphere, which is also equal to 101.3 kilopascals.

These precise conditions allow scientists to compare the behavior of gases under controlled circumstances. It's important to memorize these values, as they are frequently used in chemistry problems involving gases.
Molar Volume
Molar volume is a key concept in understanding how gases occupy space. It refers to the volume that one mole of any gas will occupy under specified conditions of temperature and pressure. For an ideal gas at STP, the molar volume is calculated to be 22.4 liters per mole.
This value is derived from the ideal gas law, represented as \(PV = nRT\), where:
  • \(P\) stands for pressure.
  • \(V\) is the volume.
  • \(n\) denotes the number of moles.
  • \(R\) is the universal gas constant.
  • \(T\) symbolizes temperature, in Kelvin.
At STP, using the universal gas constant of 0.0821 L atm/mol K, we find that for one mole of gas, the volume \( V \) becomes 22.4 liters. This constant is useful when converting between the number of moles of a gas and the volume it occupies under standard conditions.
Universal Gas Constant
The universal gas constant, denoted as \(R\), is integral to the ideal gas law formula, \(PV = nRT\). It serves as a bridging constant that relates the conditions of gases to a standard context.
Depending on the units used for pressure and volume, \(R\) may have different numerical values.
Here are some commonly used values:
  • When pressure is in atmospheres and volume in liters, \(R = 0.0821\) L atm/mol K.
  • When pressure is in kilopascals and volume in liters, \(R = 8.314\) L kPa/mol K.
  • If you measure pressure in bars, \(R\) takes the value of 0.0831 L bar/mol K.
These different values ensure we can apply the ideal gas law using whatever pressure units we are working with.
Understanding the correct value of \(R\) for your units is crucial for accurately applying the ideal gas equation.

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Most popular questions from this chapter

Consider the following gases, all at STP: \(\mathrm{Ne}, \mathrm{SF}_{6}, \mathrm{~N}_{2}, \mathrm{CH}_{4}\). (a) Which gas is most likely to depart from the assumption of the kinetic- molecular theory that says there are no attractive or repulsive forces between molecules? (b) Which one is closest to an ideal gas in its behavior? (c) Which one has the highest root-mean-square molecular speed at a given temperature? (d) Which one has the highest total molecular volume relative to the space occupied by the gas? (e) Which has the highest average kinetic- molecular energy? (f) Which one would effuse more rapidly than \(\mathrm{N}_{2} ?\) (g) Which one would have the largest van der Waals \(b\) parameter?

Ammonia and hydrogen chloride react to form solid ammonium chloride: $$\mathrm{NH}_{3}(g)+\mathrm{HCl}(g) \longrightarrow \mathrm{NH}_{4} \mathrm{Cl}(s)$$ Two \(2.00-\mathrm{L}\) flasks at \(25^{\circ} \mathrm{C}\) are connected by a valve, as shown in the drawing. One flask contains \(5.00 \mathrm{~g}\) of \(\mathrm{NH}_{3}(g),\) and the other contains \(5.00 \mathrm{~g}\) of \(\mathrm{HCl}(g) .\) When the valve is opened, the gases react until one is completely consumed. (a) Which gas will remain in the system after the reaction is complete? (b) What will be the final pressure of the system after the reaction is complete? (Neglect the volume of the ammonium chloride formed.) (c) What mass of ammonium chloride will be formed?

A deep-sea diver uses a gas cylinder with a volume of \(10.0 \mathrm{~L}\) and a content of \(51.2 \mathrm{~g}\) of \(\mathrm{O}_{2}\) and \(32.6 \mathrm{~g}\) of He. Calculate the partial pressure of each gas and the total pressure if the temperature of the gas is \(19^{\circ} \mathrm{C}\).

Both Jacques Charles and Joseph Louis Guy-Lussac were avid balloonists. In his original flight in 1783 , Jacques Charles used a balloon that contained approximately \(31,150 \mathrm{~L}\) of \(\mathrm{H}_{2}\). He generated the \(\mathrm{H}_{2}\) using the reaction between iron and hydrochloric acid: $$\mathrm{Fe}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{FeCl}_{2}(a q)+\mathrm{H}_{2}(g)$$ How many kilograms of iron were needed to produce this volume of \(\mathrm{H}_{2}\) if the temperature was \(22{ }^{\circ} \mathrm{C} ?\)

Which statement concerning the van der Waals constants \(a\) and \(b\) is true? (a) The magnitude of \(a\) relates to molecular volume, whereas \(b\) relates to attractions between molecules. (b) The magnitude of \(a\) relates to attractions between molecules, whereas \(b\) relates to molecular volume. (c) The magnitudes of \(a\) and \(b\) depend on pressure. (d) The magnitudes of \(a\) and \(b\) depend on temperature.
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