91影视

A tank in a room at \(19^{\circ} \mathrm{C}\) is initially open to the atmosphere on a day when the barometric pressure is 102 kPa. A block of dry ice (solid \(\mathrm{CO}_{2}\) ) with a mass of \(15.7 \mathrm{kg}\) is dropped into the tank, which is then sealed. The reading on the tank pressure gauge initially rises very quickly, then much more slowly, eventually reaching a value of 3.27 MPa. Assume \(T_{\text {final }}=19^{\circ} \mathrm{C}\) (a) How many moles of air were in the tank initially? Neglect the volume occupied by \(\mathrm{CO}_{2}\) in the solid state, and assume that a negligible amount of \(\mathrm{CO}_{2}\) escapes prior to the sealing of the tank. (b) Estimate the percentage error made by neglecting the volume of the block of dry ice placed in the tank. (The specific gravity of solid carbon dioxide is approximately 1.56 .) (c) What is the final density (g/L) of the gas in the tank? (d) Explain the observed variation of pressure with time. More specifically, what is happening in the tank during the initial rapid pressure increase and during the later slow pressure increase?

Step by step solution

01

Calculating initial moles of air in the tank

We can use the ideal gas equation \( P V= n R T \) to find the initial moles of air in the tank. First, convert the pressure P (102 kPa) into Pa by multiplying by 1,000; convert the temperature T (19掳C) into Kelvin by adding 273.15. Considering the initial volume V of the tank, we can rearrange the equation to \( n_{air} = \frac{PV}{RT} \) where R (8.314 J/ mol路K) is the gas constant.

02

Calculating the moles and volume of CO鈧�

Calculate the moles of CO鈧� using the mass and the molar mass of CO鈧� (44 g/mol): \( n_{CO2} = \frac{mass_{CO2}}{MolarMass_{CO2}} \). The volume of CO鈧� can be estimated using the specific gravity and the mass of CO鈧�: \( V_{CO2} = \frac{mass_{CO2}}{density_{CO2}} \) where density of CO鈧� is obtained by multiplying the specific gravity by the density of water (1,000 kg/m鲁). The percentage error due to neglecting this volume is then \( \frac{V_{CO2}}{V_{tank}} * 100\% \)

03

Calculating the final density of the gas in the tank

Total mole of gas in the tank is given by \( n_{total} = n_{air} + n_{CO2} \). Then, applying the ideal gas law again in the final situation and solving for density (蟻), we find \( 蟻 = \frac{n_{total} * MolarMass_{air}}{V_{tank}} \) where MolarMass_air is the molar mass of air (28.97 g/mol).

04

Interpreting the variation of pressure with time

The initial rapid pressure increase in the tank is due to the sublimation of solid CO鈧� into gas, increasing the number of gas particles and hence the pressure. The later slow pressure increase is due to the warm room temperature causing further expansion of the gas particles, thus gradually increasing the pressure.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pressure Variation

In the exercise, we observe two distinct phases regarding the change in pressure inside the tank. The initial rapid pressure increase occurs when the dry ice, which is solid carbon dioxide (CO鈧�), is introduced into the tank and begins to sublimate.
This means that the CO鈧� transitions directly from a solid state to a gaseous state, significantly adding to the number of gas particles in the tank and consequently leading to a quick rise in pressure.

After the initial burst of pressure, the rate of increase slows down. This is because, once the tank is sealed, the additional pressure changes depend more on temperature than on the sublimation process.
The room temperature, which remains consistent, causes a gradual expansion of the gas particles.
Gas particles move faster and further apart as their kinetic energy increases with temperature, leading to a slow but steady increase in pressure over time.

Key points to understand pressure variation include:

• The role of sublimation in rapidly increasing pressure by adding gas particles.
• The effect of temperature on slowly increasing pressure by expanding gas particles.
• Understanding pressure dynamics is essential for applications involving enclosed environments where temperature and phase changes occur.

Moles Calculation

The Ideal Gas Law, represented by the formula \( PV = nRT \), is a powerful tool for determining the number of moles of a gas.
In this exercise, to find the initial moles of air in the tank, we convert given atmospheric pressure from kilopascals to pascals and temperature from Celsius to Kelvin.
The number of moles \( n \) is then given by rearranging the Ideal Gas Law to \( n = \frac{PV}{RT} \).

The gas constant \( R \) is a universal value that makes these calculations consistent, typically using \( 8.314 \, \text{J/mol}\cdot\text{K} \).
This relationship shows the direct proportionality between the amount of gas (in moles) and the product of pressure and volume over the temperature.

Mole calculations help us:

• Predict how much product will form in a chemical reaction.
• Understand the relationship between pressure, volume, and temperature in a given setting.
• Ensure accuracy in scientific measurements and industrial applications involving gases.

Gas Density

Gas density refers to the mass of gas per unit volume and is a critical parameter in understanding how gases will behave under different conditions.
In this exercise, after determining the total number of gas moles in the tank, we apply the Ideal Gas Law to ascertain the gas density.
The final density is calculated as \( \rho = \frac{n_{total} \times \text{MolarMass}_{\text{air}}}{V_{\text{tank}}} \), where the molar mass of air is approximately \( 28.97 \, \text{g/mol} \).

This formula highlights that density is dependent on both the total amount of gas added and its molar mass.
Gas density is directly proportional to pressure and inversely proportional to volume and temperature when we consider a closed system.

Understanding gas density is crucial because it affects:

• The buoyancy behavior in different fluids, impacting things like balloon flights and airborne toxin dispersion.
• The sound speed in gases since denser gases conduct sound more slowly.
• Calculations needed in thermodynamics and fluid mechanics for designing efficient systems.

CO2 Sublimation

Sublimation is the process in which a solid transitions directly into a gas without passing through a liquid phase.
One of the most well-known sublimating substances is Carbon Dioxide (CO鈧�) - commonly referred to as dry ice.
When dry ice is placed into an open container at room temperature, it will sublimate entirely into gaseous CO鈧�.

During sublimation, there is a rapid increase in the number of molecules in the gaseous state, translating into an increase in pressure inside a confined space.
For CO鈧�, sublimation occurs at temperatures below its triple point, where the solid ice skips the liquid state and directly turns into a gas.

Here is why understanding CO鈧� sublimation is important:

• It helps in refrigeration processes, where cooling is achieved by removing heat via the sublimating gas.
• It requires caution in handling because sublimated CO鈧� can cause pressure buildup in sealed containers.
• It's useful in industries where rapid transitions from solid to gas are necessary, such as cleaning objects without residue.