Question

Gas \(\mathrm{A}\) is at \(30{ }^{\circ} \mathrm{C}\) and gas \(\mathrm{B}\) is at \(20^{\circ} \mathrm{C}\). Both gases are at 1 atmosphere. What is the ratio of the volume of 1 mole of gas A to 1 mole of gas B? 1\. \(1: 1\) 2\. \(2: 3\) 3\. \(3: 2\) 4\. \(303: 293\)

Short answer

4. \(303: 293\)

Step-by-step solution

Convert temperatures to Kelvin

Before applying the Ideal Gas Law, convert the temperatures of gases A and B from Celsius to Kelvin. To convert from Celsius to Kelvin, simply add 273.15 to the Celsius value. Gas A temperature: \(T_A = 30^{\circ}C + 273.15 = 303.15K\) Gas B temperature: \(T_B = 20^{\circ}C + 273.15 = 293.15K\)

Setup proportion using Ideal Gas Law

Since we have equal moles and pressure for both gases, the proportion between their volumes can be derived from the Ideal Gas Law: \(\frac{V_A}{V_B} = \frac{n_A RT_A}{n_B RT_B}\) Here, we can cancel out \(n_A\), \(n_B\), and \(R\) since both have the same values: \(\frac{V_A}{V_B} = \frac{T_A}{T_B}\)

Calculate the ratio of the volumes

Plug in the temperatures of both gases in Kelvin and calculate the ratio: \(\frac{V_A}{V_B} = \frac{303.15K}{293.15K} = 1.034\) Since the ratio is close to 1.03, we can approximate it as: \(\frac{V_A}{V_B} \approx \frac{303}{293}\)

Find the answer among the given options

Looking at the answer choices, the ratio we found (\(\frac{303}{293}\)) corresponds to option 4. Answer: 4. \(303: 293\)

Key concepts

Ideal Gas Law

The Ideal Gas Law is an equation that relates the pressure, volume, temperature, and amount (in moles) of an ideal gas, allowing us to predict the behavior of a gas under certain conditions. It's represented by the formula:

\[ PV = nRT \]
Where:

• \( P \) is the pressure of the gas,
• \( V \) is the volume of the gas,
• \( n \) is the amount of substance in moles,
• \( R \) is the ideal gas constant, and
• \( T \) is the temperature in Kelvin.

The law assumes that all collisions between gas molecules are perfectly elastic and that gas molecules occupy no volume. When comparing the volumes of gases under the same pressure and temperature conditions, as in the given exercise, we can use this law to derive a simple proportion (provided the number of moles is consistent) between their volumes. This concept is crucial in understanding how temperature and pressure affect gas behavior in enclosed spaces like cylinders and balloons.

Mol Volume Ratio

The mol volume ratio in gaseous systems is a way of directly comparing the space that different moles of gas occupy under certain conditions. Given the same type of gas under identical conditions, the volume is directly proportional to the number of moles. This stems from Avogadro's hypothesis, indicating that equal volumes of gases, at the same temperature and pressure, contain the same number of molecules. Thus, when you have gases under similar conditions of temperature and pressure, you can simplify the Ideal Gas Law.

In the context of the LSAT logic games or exercises involving gas properties, understanding the mole volume ratio enables you to solve for unknown volumes of gases, provided you have one point of reference. Here, you saw that when the number of moles and constant values are equal, the volumes were directly proportional to the respective temperatures in Kelvin, leading to the volume ratio.

Temperature Conversion to Kelvin

Temperature conversion to Kelvin is a fundamental step when working with gas laws because the Ideal Gas Law requires temperature to be measured in Kelvin (K). The Kelvin scale is an absolute temperature scale where 0 K represents the absence of thermal energy.

To convert Celsius to Kelvin, you add 273.15 to the Celsius temperature:
\[ T(K) = T(^{\text{\textdegree}}C) + 273.15 \]
Applying this conversion ensures that all calculations concerning gas properties are consistent with the absolute temperature scale, as seen in the exercise. In solving gas problems, this step is crucial because accidentally using degrees Celsius could lead to incorrect ratios or predictions on gas behavior. Always check your temperature units to maintain accuracy in the calculations.