Question

During a deep breath, our lungs expand about \(2.0 \mathrm{~L}\) against an external pressure of \(101.3 \mathrm{kPa}\). How much work is involved in this process (in J)?

Short answer

The work done during this process is \(-202.6 \mathrm{J}\).

Step-by-step solution

Write the formula for work done

We will use the formula of mechanical work done by a gas in this situation. Work done (W) by a gas during the expansion is given by: W = -P × ΔV where P = external pressure, ΔV = change in volume. We have a negative sign because the work done by the gas on the surroundings is negative as the gas is expanding.

Substitute the given values

Now we will substitute the given values in the formula: P = 101.3 kPa = 101.3 × 10^3 Pa (converting kPa to Pa) ΔV = 2.0 L = 2.0 × 10^-3 m³ (converting L to m³) W = -101.3 × 10^3 Pa × 2.0 × 10^-3 m³

Calculate the work done

Now, find the work done by multiplying the given values. W = -101.3 × 10^3 × 2.0 × 10^-3 W = -202.6 J The work done during this process is -202.6 J. The negative sign indicates that the work is done by the body, expanding the lungs against the external pressure.

Key concepts

Work Done by Gas

When gas within an enclosed space is made to expand, it performs work against the surrounding environment. This is known as "work done by gas." Essentially, any time a gas pushes against its container, such as when you inflate a balloon or take a deep breath, it does work. The amount of work done is determined by the formula:
\[ W = -P \times \Delta V \]
In this formula, \(W\) represents the work done, \(P\) is the external pressure exerted upon the gas, and \(\Delta V\) is the change in volume of the gas. Notably, there is a negative sign in the equation. This negative sign indicates that the work executed by the gas is done against the environment, meaning the surroundings resist the expansion.
Bullets to help:

• Work is done by the gas when it expands against an opposing pressure.
• The negative sign in the formula indicates the direction of work (outward from the system).
• This type of work is important in fields like engineering and biology, such as measuring lung function or engine efficiency.

Pressure-Volume Work

Pressure-volume work refers to the work done as a result of the pressure acting over a volume change in an elastic container, like our lungs or a piston. Whenever there is a change in volume while a constant pressure is applied, pressure-volume work comes into play.
The formula that governs this is essentially the same:\[ W = -P \times \Delta V \]With pressure-volume work, understanding the relationship between pressure and volume, known as Boyle's Law, can be very helpful:

\(P_1 \times V_1 = P_2 \times V_2\)

This law underscores that pressure and volume are inversely proportional, meaning if volume goes up, pressure must decrease to maintain the same amount of energy (and vice-versa).
For further comprehension:

• Pressure-volume work is essential in many natural and engineered systems.
• Understanding it assists in designing more efficient machines and devices, like motors or lung ventilators.
• It illustrates the mechanical energy transferred in processes involving gas expansion or compression.

Conversion of Units

To accurately calculate and understand processes like work done by gas, you often need to convert units to a consistent system. For instance, pressures are often expressed in kilopascals (kPa) while volumes might be in liters (L). However, our mathematical formulas require uniform units, typically Pascals (Pa) for pressure and cubic meters (m³) for volume, in the SI unit system.
Here's a guide on how to carry out these conversions:

• To convert kPa to Pa: Multiply by \(10^3\). For example, \(101.3\, \text{kPa} = 101,300\, \text{Pa}\).
• To convert liters to cubic meters: Multiply by \(10^{-3}\). For example, \(2.0\, \text{L} = 2.0 \times 10^{-3}\, \text{m}^3\).

Using consistent units

• Ensures accuracy in calculations and helps in understanding the relationship between different physical quantities.
• Is crucial for scientific research and standardized industrial practices.

This practice of unit consistency is vital to avoid errors and misinterpretations.