Problem 29 - Chemistry: The Central Science

Question

(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 1 atm pressure. (d) If you measure pressure in bars instead of atmospheres, calculate the corresponding value of \(R\) in L-bar/mol-K.

Step-by-Step Solution

Answer

(a) STP refers to standard temperature and pressure, which are \(0^{\circ} \mathrm{C}\) or 273.15 K and 1 atm. (b) The molar volume of an ideal gas at STP is approximately 22.4 L/mol. (c) At \(25^{\circ} \mathrm{C}\) and 1 atm, the molar volume of an ideal gas is around 24.5 L/mol. (d) The gas constant R in L-bar/mol-K is approximately 0.08314 L-bar/mol-K.

Step 1: (a) STP Conditions

The abbreviation STP stands for standard temperature and pressure. These conditions are defined as a temperature of \(0^{\circ} \mathrm{C}\) or 273.15 K and a pressure of 1 atmosphere (1 atm).

Step 2: (b) Molar volume at STP

To find the molar volume of an ideal gas at STP, we can use the ideal gas law: \(PV = nRT\) Where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. Since we want to know the molar volume (i.e., volume per 1 mole), we can express the ideal gas law as: \(V = \dfrac{RT}{P}\) We know standard conditions are \(T = 273.15 \mathrm{K}\) and \(P = 1 \mathrm{atm}\). The value of the gas constant R, when pressure is measured in atmospheres, is approximately \(0.0821 \mathrm{L\cdot atm/(mol\cdot K)}\). Now, let's substitute these values into our equation: \(V = \dfrac{(0.0821 \mathrm{L\cdot atm/(mol\cdot K)})(273.15 \mathrm{K})}{1 \mathrm{atm}}\) \(V \approx 22.4 \mathrm{L/mol}\) So, the molar volume of an ideal gas at STP is approximately 22.4 L/mol.

Step 3: (c) Molar volume at \(25^{\circ} \mathrm{C}\) and 1 atm

To find the molar volume of an ideal gas at \(25^{\circ} \mathrm{C}\) and 1 atm, we can use the same equation as in part (b): \(V = \dfrac{RT}{P}\) This time, we have \(T = 25^{\circ} \mathrm{C}\) or 298.15 K, and \(P = 1 \mathrm{atm}\). The gas constant R remains the same with \(0.0821 \mathrm{L\cdot atm/(mol\cdot K)}\). Substituting these values into the formula, we get: \(V = \dfrac{(0.0821 \mathrm{L\cdot atm/(mol\cdot K)})(298.15 \mathrm{K})}{1 \mathrm{atm}}\) \(V \approx 24.5 \mathrm{L/mol}\) The molar volume of an ideal gas at \(25^{\circ} \mathrm{C}\) and 1 atm is approximately 24.5 L/mol.

Step 4: (d) Gas constant value in L-bar/mol-K

Currently, we have the gas constant value in L-atm/mol-K. We want to convert it to L-bar/mol-K. To do this, we can use the conversion factor between bars and atmospheres. Since 1 atm is equivalent to 1.01325 bars, we can write: \(R = 0.0821 \mathrm{L\cdot atm/(mol\cdot K)} \times \dfrac{1 \mathrm{atm}}{1.01325 \mathrm{bar}}\) \(R \approx 0.08314 \mathrm{L\cdot bar/(mol\cdot K)}\) The corresponding value of the gas constant R in L-bar/mol-K is approximately 0.08314 L-bar/mol-K.

Key Concepts

STP (Standard Temperature and Pressure)

STP, or Standard Temperature and Pressure, is a term used to define a set of specific conditions for scientific experiments and calculations involving gases. At STP, the temperature is exactly 0 degrees Celsius (or 273.15 Kelvin) and the pressure is exactly 1 atmosphere. These conditions are significant because they provide a common reference point for comparing the results of scientific experiments related to gases. STP is especially important in the study of gases because it simplifies the calculations of volume, pressure, and temperature relationships. By knowing the conditions at STP, we can make accurate predictions and calculations about gas behaviors.

Molar Volume

Molar volume refers to the volume occupied by one mole of a substance at a given set of conditions, typically measured in liters per mole (L/mol). For an ideal gas at STP, the molar volume is approximately 22.4 L/mol. This value is derived using the Ideal Gas Law, which relates pressure, volume, temperature, and the number of moles.
To find the molar volume of an ideal gas at STP, we use the equation:

\[ V = \dfrac{RT}{P} \]where:

• \( V \) is the volume,
• \( R \) is the gas constant,
• \( T \) is the temperature in Kelvin,
• \( P \) is the pressure in atm.

This simple formula shows that at STP, all ideal gases will occupy 22.4 liters per mole, making it a valuable tool in chemistry for conversions and calculations.

Gas Constant R

The gas constant, often denoted as \( R \), is a key component of the Ideal Gas Law equation \( PV = nRT \). It acts as a bridge connecting energy scales and interpreting gas behaviors under different conditions. The value of \( R \) depends on the units used for pressure and volume. Commonly, you will see the value used in calculations as 0.0821 L·atm/mol·K when pressure is in atmospheres and volume is in liters.To convert \( R \) to other units, such as L·bar/mol·K, you can use the conversion factor between atmospheres and bars, given that 1 atm equals 1.01325 bars. This conversion changes the value to approximately 0.08314 L·bar/mol·K. Understanding how to convert and use \( R \) in calculations allows for greater flexibility when solving gas-related problems under varying conditions.

Unit Conversion

Unit conversion is an essential skill in chemistry, particularly when dealing with different measurement systems. In the context of the ideal gas law, converting between units like atmospheres and bars is crucial for accurate calculations. For example, you might need to convert pressure from atm to bars by using the conversion: 1 atm = 1.01325 bars. This ensures that variables within the Ideal Gas Law equation are consistent, which is necessary for the equation to hold true. Another common conversion involves temperature scales, converting Celsius to Kelvin by adding 273.15 to Celsius degrees, as the Kelvin scale is used in gas law calculations. By mastering unit conversions, you can efficiently transition between different measurement systems, enhancing your ability to solve diverse problems reliably and accurately.