Problem 47 - General Chemistry: Principles and Modern Applications

Question

Calculate the quantity of work associated with a \(3.5 \mathrm{L}\) expansion of a gas \((\Delta V)\) against a pressure of \(748 \space\mathrm{mmHg}\) in the units (a) atm \(\mathrm{L} ;\) (b) joules (J); (c) calories (cal).

Step-by-Step Solution

Answer

Based on the step-by-step solution, the calculated work done by the gas against the pressure is \( -2.614 \) atm L, \( -264.3 \) J and \( -63.19 \) cal.

Step 1: Convert pressure to atm

To calculate work in atm L, first convert pressure from mmHg to atm. Use the conversion factor 1 atm = 760 mmHg.

Step 2: Calculate Work (atm L)

Calculate work by plugging the values of pressure (in atm) and the volume change (in L) into the formula for work, \( W = -P \Delta V \).

Step 3: Convert Work to Joules

To calculate work in joules (J), convert the result from Step 2, which is in atm L, to J. Use the conversion factor 1 atm L = 101.3 J.

Step 4: Convert Work to Calories

To calculate work in calories (cal), convert the result from Step 2, which is in atm L, to J first (using the conversion factor from Step 3) and then convert J to cal, using the conversion factor 1 cal = 4.184 J.

Key Concepts

Work Calculation

Work calculation in thermodynamics involves determining the energy required or released due to the expansion or compression of a gas. The basic formula is:

• Work \( W = -P \Delta V \)

Here, \( P \) is pressure and \( \Delta V \) is the change in volume. The negative sign indicates that work is done by the system when it expands.
Understanding this formula helps in calculating the work associated with processes that involve gas, like inflating a balloon or a piston in an engine.
For example, if a gas in a balloon expands by 3.5 L against a certain pressure, we use this formula to find out how much work was done by the gas as it pushed back the surrounding pressure.

Pressure Conversion

Pressure conversion is a necessary step when dealing with pressure measurements in different units. In our problem, pressure is given in mmHg (millimeters of mercury) but needs to be converted to atm (atmospheres) for use in the work calculation formula.

This conversion uses the fact that:

• 1 atm = 760 mmHg

So to convert from mmHg to atm, divide the given pressure in mmHg by 760.
Once pressure is in atm, it can be easily plugged into various equations in thermodynamics to calculate work and other properties.

Energy Units

In thermodynamics, work and energy can be expressed in different units, commonly in atm L, joules, and calories. Each unit serves a different purpose:

• atm L: Used mainly for calculations in chemistry related to gas laws, as it naturally incorporates gas volume and pressure.
• Joules: The SI unit for energy. It is widely used in physics and engineering. To convert work from atm L to joules, use the conversion factor 1 atm L = 101.3 J.
• Calories: Commonly used in heat-related calculations, especially in chemistry and biochemistry. To convert energy from joules to calories, use the conversion factor 1 cal = 4.184 J.

Knowing these conversion factors allows you to interchange energy units as needed, depending on the context or field of study.

Gas Expansion

Gas expansion occurs when a gas increases its volume against an external pressure. It is a common process studied in thermodynamics. When a gas expands, it does work on its surroundings by pushing against them.

Understanding gas expansion helps analyze processes like balloon inflating or engines where gases do work.

• Important factors:
• Volume change \( \Delta V \) represents the amount of expansion.
• External pressure \( P \) against which the gas expands.

By applying the formula \( W = -P \Delta V \), you can quantify the work done during gas expansion, giving insights into the energy exchange in numerous practical applications.