Problem 119 - Chemistry and Chemical Reactivity

Question

Sublimation of \(1.0 \mathrm{g}\) of dry ice, \(\mathrm{CO}_{2}(\mathrm{s}),\) forms \(0.36 \mathrm{L}\) of \(\mathrm{CO}_{2}(\mathrm{g})\left(\mathrm{at}-78^{\circ} \mathrm{C} \text { and } 1.01 \times 10^{5} \mathrm{Pa}\right)\) The expanding gas can do work on the surroundings (Figure 5.8 ). Calculate the amount of work done on the surroundings.

Step-by-Step Solution

Answer

The work done by the gas on the surroundings is \(-36.36 \text{J}\).

Step 1: Understand the Formula for Work Done by a Gas

The work done by a gas when it expands can be calculated using the formula \[ w = -P \Delta V \]where \( w \) is the work done, \( P \) is the external pressure, and \( \Delta V \) is the change in volume.

Step 2: Identify Given Values

We are given that the gas occupies a volume of \( 0.36 \mathrm{L} \) at a pressure of \( 1.01 \times 10^{5} \mathrm{Pa} \) (or \( 1.01 \times 10^{5} \mathrm{N/m^2} \)). Remember that work is performed when volume changes from initial to final volume, here initial volume was essentially 0 as it was solid \(\mathrm{CO}_{2}(\mathrm{s}) \).

Step 3: Convert Volume from Liters to Cubic Meters

Since pressure is given in pascals, we need to ensure that all units are compatible. Convert the volume from liters to cubic meters using the conversion factor:\[ 1 \text{L} = 1 \times 10^{-3} \text{m}^3 \]Thus,\[ 0.36 \text{L} = 0.36 \times 10^{-3} \text{m}^3 = 0.00036 \text{m}^3 \]

Step 4: Calculate Work Done Using the Formula

Substitute the known values into the formula:\[ w = -P \Delta V = -(1.01 \times 10^5 \text{Pa}) \times (0.00036 \text{m}^3)\]Complete the multiplication:\[ w = -(1.01 \times 10^5) \times 0.00036 = -36.36 \text{J} \]The work done by the gas on the surroundings is \(-36.36 \text{J} \). The negative sign indicates work done by the system on its surroundings.

Key Concepts

Sublimation

Sublimation is a fascinating thermodynamic process where a substance transitions directly from a solid to a gas without passing through the liquid state. This is what occurs with dry ice or solid carbon dioxide under normal atmospheric conditions. It might seem surprising that a solid can bypass the liquid phase, but sublimation occurs because the molecules in some substances gain enough energy to break away from their neighbors and enter the gas phase.
- For sublimation to take place, a certain amount of energy is needed, known as enthalpy of sublimation. - This process is endothermic, meaning it requires heat to be absorbed from the surroundings. Dry ice sublimates at temperatures of extstyle{-78^{ ext{°}} ext{C}}), which is why you can sometimes see it turn into thick fog-like gas as it comes into contact with warmer air. This fog is essentially water vapor from the air, condensing upon contact with the cold carbon dioxide gas diffusing into the atmosphere.

Work Done by Gas

When we talk about the work done by a gas, we are exploring one of the fundamental concepts of thermodynamics. As a gas expands, it can exert force on its surroundings, displacing them. This is termed as doing work.

- The work done by a gas during expansion or compression can be expressed mathematically by the equation \[ w = -P \Delta V \]- Here, "\(w\)" represents the work done by the system, "\(P\)" is the external pressure, and "\(\Delta V\)" stands for the change in volume from the initial to the final state. In our exercise about dry ice, since the carbon dioxide gas expanded as it sublimated, it performed work against the atmospheric pressure. The negative sign in the equation denotes that the work is done by the system on its surroundings. This is an important aspect to remember, as it aligns with the convention used in thermodynamics - The system loses energy when it does work on the surroundings, which is indicated by the negative sign.

Pressure-Volume Work

In thermodynamics, pressure-volume work refers to the work done by or on a system as it undergoes a volume change under constant pressure. This is a special case of work known as "plane work" or "PdV work," crucial in processes where the volume is changing.
- The formula used to calculate pressure-volume work is \[ w = -P \Delta V \] - This type of work is common in processes involving gas, such as engines, refrigerators, and indeed, the sublimation of dry ice.The negative sign preceding the product of pressure and change in volume reminds us that the system is doing work on the surroundings. When the gas expands, the volume increases, leading to negative work done in thermodynamic terms. This signifies that energy is being transferred from the system to its environment.
- Understanding this helps us to interpret how mechanical work is a manifestation of energy loss from the system as volume expands.

Ideal Gas Law

The Ideal Gas Law is a simple yet powerful equation that provides insight into the relationship between pressure, volume, temperature, and number of moles of a gas. The equation is expressed as \[ PV = nRT \] where "\(P\)" stands for pressure, "\(V\)" is volume, "\(n\)" is the amount of substance in moles, "\(R\)" is the universal gas constant, and "\(T\)" is temperature in Kelvin.Though the gas in our example is carbon dioxide, sublimated from dry ice, the behavior can be approximated by the Ideal Gas Law for calculations.

- Understanding how the variables interact in this equation allows us to predict how one variable will change when another is altered. - For instance, in a constant-volume process, an increase in temperature results in an increase in pressure, a concept widely used in thermodynamic problem-solving.While the Ideal Gas Law assumes a perfect behavior of gases with no interactions between molecules and no volume occupied by the gas molecules themselves, it serves as an excellent approximation in many practical scenarios, including our exercise. Utilizing this law, we arrive at predictions and understandings relevant to processes like sublimation and gas expansion.