Question

The volume of a gas at \(20 \mathrm{C}\) is \(200 \mathrm{ml}\). If the temperature is reduced to \(-20^{\circ} \mathrm{C}\) at constant pressure, its volume will be. (A) \(172.6 \mathrm{~m} 1\) (B) \(17.26 \mathrm{ml}\) (C) \(19.27 \mathrm{ml}\) (D) \(192.7 \mathrm{ml}\)

Short answer

The correct answer is (A) \(172.6 \mathrm{ml}\) as the final volume of the gas when the temperature is reduced to \(-20^{\circ} \mathrm{C}\) at constant pressure.

Step-by-step solution

Convert temperatures to Kelvin scale

In order to work with the gas law equations, we must convert the Celsius temperatures to Kelvin. The Kelvin scale is found by adding 273.15 to the Celsius temperature: Initial temperature in Kelvin: \(T_1 = 20 ^\circ \mathrm{C} + 273.15 = 293.15 \mathrm{K}\) Final temperature in Kelvin: \(T_2 = -20 ^\circ \mathrm{C} + 273.15 = 253.15 \mathrm{K}\)

Apply Charles's Law

Charles's Law states that the volume of a gas is directly proportional to its temperature when at constant pressure. This relationship can be written as: \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\) Now, we can plug in the known values and solve for \(V_2\): \(V_1\) is given as 200 ml, \(T_1\) is 293.15 K, and \(T_2\) is 253.15 K. \(\frac{200 \mathrm{ml}}{293.15 \mathrm{K}} = \frac{V_2}{253.15 \mathrm{K}}\)

Solve for the new volume \(V_2\)

Now we need to solve the equation for the final volume, \(V_2\): \(V_2 = \frac{200 \mathrm{ml} \cdot 253.15 \mathrm{K}}{293.15 \mathrm{K}}\) Calculating this expression, we find: \(V_2 = 172.6179 \mathrm{ml}\) Since the problem's answers are rounded to two decimal places, we will also round our answer to two decimal places: \(V_2 ≈ 172.62 \mathrm{ml}\) Comparing our answer with the given options, we find that our answer is approximately equal to option (A) \(172.6\) ml. Therefore, the correct answer is: (A) \(172.6 \mathrm{ml}\)

Key concepts

Gas Laws

Gas laws are fundamental principles that describe the behavior of gases under different conditions. These laws are essential in understanding how gases react when changes in pressure, temperature, or volume occur.
One of the most well-known gas laws is Charles's Law, which directly involves the relationship between the volume and temperature of a gas. It states that the volume of gas increases with an increase in temperature, provided the pressure remains constant.
Other important gas laws include Boyle's Law, which relates pressure and volume, and Avogadro's Law, which connects the volume of a gas to the number of moles present. By understanding and applying these gas laws, we can predict and calculate the behavior of gas in different scenarios, aiding in various scientific and real-world applications.

Temperature Conversion

When working with gas laws, it's crucial to use an absolute temperature scale like Kelvin. The Celsius scale, while familiar, starts at water's freezing point, making it non-absolute. The Kelvin scale, however, starts at absolute zero, where particle motion theoretically ceases.
Converting Celsius to Kelvin is straightforward. Simply add 273.15 to the Celsius temperature. For example, to convert 20°C, you calculate 20 + 273.15 = 293.15 K.
This conversion is necessary because gas laws, like Charles's Law, rely on the direct proportionality of temperature and volume, which only holds true when temperature is measured from the absolute zero point. By using Kelvin, we ensure accurate calculations and predictions in gas law problems.

Volume and Temperature Relationship

The relationship between volume and temperature for a gas at constant pressure is crucial in gas behavior studies. Known as Charles's Law, this principle asserts that the volume of a gas varies directly with its temperature in Kelvin.
Mathematically, this is represented by the equation \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \), where \( V_1 \) and \( V_2 \) are the initial and final volumes, and \( T_1 \) and \( T_2 \) are the initial and final temperatures in Kelvin, respectively.
As the temperature increases, so does the energy of the gas particles, causing them to move more vigorously and thus occupy a larger volume. Conversely, if the temperature decreases, the movement of particles slows, and the gas volume decreases. This relationship is fundamental in predicting gas behavior in applications ranging from meteorology to medicine.

Kelvin Scale

The Kelvin scale is an absolute temperature scale pivotal in scientific measurements, especially in gas laws. It is named after Lord Kelvin, who established it to provide a scale where zero represents a state of "absolute" no thermal energy.
Unlike the Celsius and Fahrenheit scales, the Kelvin scale does not have negative numbers, making mathematical calculations involving temperature more straightforward and reliable.
Every unit of temperature on the Kelvin scale corresponds exactly to one degree on the Celsius scale, but it starts at absolute zero (0 K), or -273.15°C. This point is where all molecular movement ceases. Using Kelvin ensures consistency and accuracy in scientific computations, particularly in gas-related equations where the laws rely on direct proportionality related to absolute temperature.