Question

A gas expands from 1 liter to 3 liter at atmospheric pressure. The work done by the gas is about (A) \(200 \mathrm{~J}\) (B) \(2 \mathrm{~J}\) (C) \(300 \mathrm{~J}\) (D) \(2 \times 10^{5} \mathrm{~J}\)

Short answer

The work done by the gas when it expands from 1 liter to 3 liters at atmospheric pressure can be calculated using the formula: \(W = P \times \Delta V\). After converting the pressure to pascals and the volume change to cubic meters, we get \(W = (101325 ~Pa) \times (0.002 ~m^3) = 202.65 ~J\). Therefore, the correct option is (A) \(200 \mathrm{~J}\).

Step-by-step solution

Convert atmospheric pressure to pascals

At standard atmospheric pressure, 1 atmosphere is equal to 101325 pascals (Pa). Therefore, we will first convert the pressure to pascals: \(1~ atm = 101325 ~Pa\)

Calculate volume change

The gas expands from 1 liter to 3 liters, so the volume change can be calculated as follows: \[\Delta V = V_{final} - V_{initial} = 3L - 1L = 2L\]

Convert liters to cubic meters

Gases usually have their volume measured in liters. However, to calculate work in the SI unit (Joule), we need to have volumes in cubic meters (m³). There are 1,000 liters in a cubic meter, so our volume change will be: \(\Delta V = 2L \times \frac{1 m^{3}}{1000 L} = 0.002 m^{3}\)

Calculate work

Now that we have the pressure in pascals and the volume change in cubic meters, we can calculate the work done by the gas using the formula: \[W = P \times \Delta V\] \[W = (101325 ~Pa) \times (0.002 ~m^3)\] \[W = 202.65 ~J\] Since the work done by the gas is about 202.65 J, the correct option is: (A) \(200 \mathrm{~J}\)

Key concepts

Atmospheric Pressure

Atmospheric pressure is the pressure exerted by the weight of air in the Earth's atmosphere. It's an important factor to consider when calculating the work done by gases. At sea level, standard atmospheric pressure is often used as a reference value, and it is equivalent to 101325 pascals (Pa). In problems involving gases, pressure is often provided in units called atmospheres (atm). This means we frequently need to convert atmospheres to pascals for calculations.

• 1 atm = 101325 Pa

Knowing how to convert between these units ensures we use the correct values in formula calculations. Atmospheric pressure effectively acts as a constant force that impacts the behavior and expansion of gases. It sets the stage for calculating the amount of work a gas can perform when it expands or compresses.

Volume Change

Volume change refers to the difference in the volume of a gas before and after expansion or compression. Understanding how to calculate volume change is crucial in finding out how much work is done by or on the gas. In this scenario, the gas expands from an initial volume of 1 liter to a final volume of 3 liters.

• Initial Volume (\(V_{initial}\)): 1 liter
• Final Volume (\(V_{final}\)): 3 liters
• Volume Change (\(\Delta V\)): 3L - 1L = 2L

Volume change is denoted as \(\Delta V\) and is a key part of the work calculation. The change signifies how much more space the gas occupies after expansion.

Work Formula Calculation

Calculating work done by a gas involves using the formula for work, which is the product of pressure and volume change. The formula is expressed as:oh\[ W = P \times \Delta V \]Where \(W\) represents the work done, \(P\) is the pressure, and \(\Delta V\) is the volume change in cubic meters. For this exercise:

• Pressure (\(P\)): 101325 Pa
• Volume Change (\(\Delta V\)): converted to cubic meters as 0.002 m³
• Work (\(W\)): 101325 Pa × 0.002 m³ = 202.65 J

This calculation helps us determine that the amount of work done by the gas during expansion is about 202.65 Joules.

Unit Conversion

Unit conversion plays a vital role when solving physics problems involving gases. The initial values often start in one unit, but require conversion to another to properly plug into formulas. For the work done by the gas problem, it is necessary to convert both pressure and volume into their respective SI units:

• Pressure: Convert atmospheric pressure from atm to Pa (1 atm = 101325 Pa)
• Volume: Convert volume from liters to cubic meters (1 L = 0.001 m³)

Effectively using unit conversion ensures precision and accuracy in physics calculations. It prevents errors and makes sure the values are consistent with standard units, allowing us to compute the work done in Joules, the accepted SI unit for energy.