Question

The enthalpy of sublimation ( solid \(\rightarrow\) gas) for dry ice (i.e., \(\mathrm{CO}_{2}\) ) is \(\Delta H_{\mathrm{sub}}^{\circ}=571 \mathrm{kJ} / \mathrm{kg}\) at \(-78.5^{\circ} \mathrm{C} .\) If \(125.0 \mathrm{J}\) of heat is transferred to a block of dry ice that is \(-78.5^{\circ} \mathrm{C},\) what volume of \(\mathrm{CO}_{2} \operatorname{gas}(d=1.98 \mathrm{g} / \mathrm{L})\) will be generated?

Short answer

The volume of \( CO_2 \) gas produced is \( 0.111 \ L \)

Step-by-step solution

Identify the known variables

The given known variables in this problem are: enthalpy of sublimation \(\Delta H_{sub}^\circ = 571 \ kJ/kg\), the mass of dry ice that is -78.5 C, \( m = 125.0 \ J \), the heat is transferred to a block of dry ice, \( q = 125.0 \ J \), the density of \( CO_2 \) gas, \( d = 1.98 \ g/L \)

Convert the joules to kJ

The given heat \( q \) is in joules, however, the given enthalpy of sublimation \(\Delta H_{sub}\) is in kJ/kg. Therefore the given heat should be converted to kJ in order to be compatible for the following calculations. \( 1 \ kJ = 1000 \ J\), therefore, \( 125.0 \ J = 125.0 / 1000 \ kJ = 0.125 \ kJ \)

Calculate the mass of \( CO_2 \)

The formula for the heat transfer in a change of state is given as \( q = m \delta H \). Solving for \( m \), the mass, gives, \( m = q / \delta H = 0.125 \ kJ / 571 \ kJ/kg = 2.19 x 10^-4 \ kg \). Since the amount required is in grams, this mass should be converted to grams. \( 1 \ kg = 1000 \ g \), so \( 2.19 x 10^-4 \ kg = 2.19 x 10^-4 x 1000 \ g = 0.219 \ g \)

Calculate the volume of \( CO_2 \) gas

Density, \( d \), is defined as mass over volume or \( d = m / V \). Solving for \( V \), the volume, gives \( V = m / d = 0.219 \ g / 1.98 \ g/L = 0.111 \ L \)

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships and conversions between heat and other forms of energy. In thermodynamics, we study how energy is transferred through systems and the effect it has on the physical properties of matter.

In the context of this exercise, thermodynamics explains how heat energy, or enthalpy, is involved when dry ice sublimates. Enthalpy of sublimation is the quantity of heat required to change a solid directly into a gas at a constant pressure. In our problem, the enthalpy of sublimation for dry ice is given as 571 kJ per kg.

• Energy Transfer: The problem involves transferring 125.0 J of heat to dry ice, causing a phase change from solid to gas. This is a key thermodynamic process because it describes how energy changes the state of matter.
• Equilibrium: The system stays at constant temperature time during sublimation since the enthalpy of sublimation dictates the energy needed to change phases at a specific temperature.

Phase Change

Phase change refers to the transition of matter from one state to another, such as solid to liquid, liquid to gas, or solid to gas (sublimation). In the context of our exercise, sublimation is the focus.

Sublimation is the process where a solid changes directly into a gas without passing through the liquid phase. For dry ice ( e.g., carbon dioxide at -78.5°C), this occurs readily at room temperature. It’s a fascinating process because it requires a specific amount of energy, known as the enthalpy of sublimation, to break the bonds in the solid and form gas molecules.

• Direct Transition: Sublimation is unique as it skips the liquid phase. This means the energy required for sublimation is generally higher compared to regular melting or boiling.
• Impact of Temperature: The provided temperature of -78.5°C is crucial. At this point, the energy allows carbon dioxide molecules to gain enough motion to transform into a gas.

Gas Density

Gas density is a measure of how much mass of a gas is contained in a given volume. It is usually expressed in terms like grams per liter (g/L). In this exercise, the density of carbon dioxide gas is given as 1.98 g/L.

Understanding gas density is essential when calculating how much gas is produced from a given mass when a solid sublimates. Around us, gas density affects how gases behave in different conditions like pressure and temperature.

• Calculation Importance: By determining the gas density, the volume of gas produced can be inferred from the known mass of dry ice after it sublimates.
• Practical Applications: In real-life scenarios, knowing the gas density helps predict how gases will spread or compact in a space, especially when dealing with gaseous substances in industries or scientific research.