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Chapter 5: Problem 13

The volume of a dry box (a closed chamber with dry nitrogen flowing through it) is \(2.0 \mathrm{m}^{3}\). The dry box is maintained at a slight positive gauge pressure of \(10 \mathrm{cm} \mathrm{H}_{2} \mathrm{O}\) and room temperature \(\left(25^{\circ} \mathrm{C}\right) .\) If the contents of the box are to be replaced every five minutes, calculate the required mass flow rate of nitrogen in \(g / \min\) by (a) direct solution of the ideal-gas equation of state and (b) conversion from standard conditions. You may assume the gas in the dry box is well mixed.

Short Answer

Expert verified
Obtain the required mass flow rate by directly applying the ideal gas law and converting the obtained value to standard conditions. The resulting value should be converted from kilograms per second (kg / s) to grams per minute (g / min) using appropriate conversion factors. The detailed steps illustrate the method for doing this.

Step by step solution

01

Determining the absolute pressure

Firstly, convert the gauge pressure from \(cmH_{2}O \) to \(\ kPa \) (kilopascal). The conversion factor is 1 \(\ cmH_{2}O \) = 0.0980665 \(\ kPa \). Hence the absolute pressure \(P\) at room temperature in \(kPa\) is \(P = 101.325 + (10\times0.0980665)\) that gives us \(P=102.256665 kPa\).
02

Using the ideal gas law for direct solution

The ideal gas law equation is \(PV = nRT\), but we want it in terms of mass flow rate \((\dot{m})\) which is \(\frac{m}{t} \), where \(t\) is the time taken to replace the gas and \(V \) the volume. We rewrite the equation in the form required: \(PV = \frac{\dot{m}}{R'}T\), where \(\dot{m}= \frac{PV}{R'T} \), \(R'\) is the specific gas constant for nitrogen (\(R'=0.2968 \, kPa.m^3/(kg.K)\)). Now, we can substitute the values of \(P\), \(V\), \(R'\) and \(T\)(temperature given) into the equation to find out \(\dot{m}\).
03

Converting to standard conditions

To calculate mass flow rate under standard conditions, we use the Standard Temperature and Pressure (STP) conditions. STP temperature \(T_{STP}=273 K\) and STP pressure \(P_{STP}=101.325 kPa\). So for STP, using \(\dot{m_{STP}}= \frac{PV}{R'T} \) equation, we substitute \(T_{STP}\) and \(P_{STP}\) while keeping the volume the same. This will provide the mass flow rate in the standard conditions.
04

Converting the units

Since the problem asked for answer in \(g/min\), the result obtained in \(Kg/sec\) has to be converted to the required units. To do so, one multiplies the results by \(60 sec / min\) to convert \(sec\) to \(min\), and by \(1000 g/Kg\) to convert \(Kg\) to \(g\).

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Key Concepts

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

Ideal Gas Law
The ideal gas law is a foundational equation in thermodynamics and physical chemistry that relates the pressure (P), volume (V), temperature (T), and amount (n) of an ideal gas, where the amount can be in terms of moles. The equation is given by:
\[ PV = nRT \]
In this context, 'R' represents the universal gas constant, which has a value dependent on the units used for pressure, volume, and temperature. When solving for the mass flow rate, or the rate at which mass enters or exits a system, the ideal gas law can be rearranged to include the mass and specific gas constant (\( R' \)).
For a given mass (\( m \)) and specific gas constant (\( R' \)), the ideal gas law becomes:
\( PV = (m/t)R'T \)
Where \( \dot{m} \) (mass flow rate) is equivalent to \( m/t \). For the calculation of mass flow rate in a particular system, such as a dry box with nitrogen gas, this form of the equation allows us to find the rate at which nitrogen must flow to maintain the specified conditions, using the pressure, volume, and temperature of the system along with the specific gas constant for nitrogen.
Standard Temperature and Pressure
Standard Temperature and Pressure (STP) are reference conditions used in the field of chemistry and physics to enable comparisons between different sets of data. STP is typically defined as a temperature of 0掳C (273.15K) and an absolute pressure of 1 atmosphere (101.325 kPa). These conditions are essential when using the ideal gas law to calculate properties such as density or volume that can vary with changes in temperature and pressure.

Calculations involving gases often require adjustments to account for deviations from STP due to the fact that real-world conditions usually differ. In the case of our mass flow rate calculation for nitrogen, converting to and from STP allows us to comprehend the changes in gas behavior under different conditions, and it is instrumental in solving for the mass flow rate under the specific conditions mentioned in the exercise.
Absolute Pressure
Pressure is essential in understanding how gases behave under different conditions, and it is a critical variable in the ideal gas law. Absolute pressure is the total pressure exerted by a gas, including the atmospheric pressure and any additional pressure applied by the gas itself. It is a sum of the gauge pressure, which can be positive or negative, and the atmospheric pressure.

For instance, in our example with the dry box, gauge pressure is given, but for accurate calculations using the ideal gas law, absolute pressure is needed. This is why the first step in the solution involves converting the gauge pressure to absolute pressure by adding it to the atmospheric pressure. The atmospheric pressure is equivalent to the pressure exerted by the Earth's atmosphere at sea level (101.325 kPa at STP). The inclusion of absolute pressure in calculations ensures accuracy when applying the ideal gas law in practical scenarios.
Gas Constant
The gas constant, often symbolized as 'R', is a physical constant that appears in several fundamental equations in the physical sciences, such as the ideal gas law. Its value depends on the units chosen for pressure, volume, and temperature, with a common value being 8.314 J/(mol路K) when pressure is in pascals and volume in cubic meters.

However, each gas also has a specific gas constant (\( R' \)), which relates the universal gas constant to the molar mass of the gas. For nitrogen, which is used in the exercise, the specific gas constant is 0.2968 kPa路m鲁/(kg路K). This particular value permits the calculation of the mass flow rate without first finding the number of moles, simplifying the process. Understanding the role of the gas constant is vital for working with gas laws and carrying out calculations related to the behavior of gases.

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Most popular questions from this chapter

The label has come off a cylinder of gas in your laboratory. You know only that one species of gas is contained in the cylinder, but you do not know whether it is hydrogen, oxygen, or nitrogen. To find out, you evacuate a 5 -liter flask, seal it and weigh it, then let gas from the cylinder flow into it until the gauge pressure equals 1.00 atm. The flask is reweighed, and the mass of the added gas is found to be 13.0g. Room temperature is \(27^{\circ} \mathrm{C}\), and barometric pressure is 1.00 atm. What is the gas?

The absolute pressure within a 35.0 -liter gas cylinder should not exceed 51.0 atm. Suppose the cylinder contains \(50.0 \mathrm{mol}\) of a gas. Use the SRK equation of state to calculate the maximum permissible cylinder temperature if the gas is (a) carbon dioxide and (b) argon. Finally, calculate the values that would be predicted by the ideal-gas equation of state.

An oxygen tank with a volume of \(2.5 \mathrm{ft}^{3}\) is kept in a room at \(50^{\circ} \mathrm{F}\). An engineer has used the idealgas equation of state to determine that if the tank is first evacuated and then charged with \(35.3 \mathrm{lb}_{\mathrm{m}}\) of pure oxygen, its rated maximum allowable working pressure (MAWP) will be attained. Operation at pressures above this value is considered unsafe. (a) What is the maximum allowable working pressure (psig) of the tank? (b) You suspect that at the conditions of the fully charged tank, ideal-gas behavior may not be a good assumption. Use the SRK equation of state to obtain a better estimate of the maximum mass of oxygen that may be charged into the tank. Did the ideal-gas assumption lead to a conservative estimate (on the safe side) or a nonconservative estimate of the amount of oxygen that could be charged? (c) Suppose the tank is charged and ruptures before the amount of oxygen calculated in Part (b) enters it. (It should have been able to withstand pressures up to four times the MAWP.) Think of at least five possible explanations for the failure of the tank below its rated pressure limit.

During your summer vacation, you plan an epic adventure trip to scale Mt. Kilimanjaro in Tanzania. Dehydration is a great danger on such a climb, and it is essential to drink enough water to make up for the amount you lose by breathing. (a) During your pre-trip physical, your physician measured the average flow rate and composition of the gas you exhaled (expired air) while performing light activity. The results were \(11.36 \mathrm{L} / \mathrm{min}\) at body temperature (37^) C) and 1 atm, 17.08 mole\% oxygen, 3.25\% carbon dioxide, 6.12 mole\% \(\mathrm{H}_{2} \mathrm{O},\) and the balance nitrogen. The ambient (inspired) air contained 1.67 mole\% water and a negligible amount of carbon dioxide. Calculate the rate of mass lost through the breathing process (kg/day) and the volume of water in liters you would have to drink per day just to replace the water lost in respiration. Consider your lungs to be a continuous steady-state system, with input streams being inspired air and water and \(\mathrm{CO}_{2}\) transferred from the blood and output streams being expired air and \(\mathrm{O}_{2}\) transferred to the blood. Assume no nitrogen is transferred to or from the blood. (b) You made the trip to Tanzania and completed the climb to Uhuru Peak, the summit of Kilimanjaro, at an altitude of 5895 meters above sea level. The ambient temperature and pressure there averaged \(-9.4^{\circ} \mathrm{C}\) and \(360 \mathrm{mm} \mathrm{Hg},\) and the air contained \(0.46 \mathrm{mole} \%\) water. The molar flow rate of your expired air was roughly the same as it had been at sea level, and the expired air contained \(14.86 \% \mathrm{O}_{2}\) \(3.80 \% \mathrm{CO}_{2},\) and \(13.20 \% \mathrm{H}_{2} \mathrm{O} .\) Calculate the rate of mass lost (g/day) through breathing and water you would have to drink (L/day) just to replace the water lost in respiration. (c) The equality of the molar flow rates of expired air at sea level and at Uhuru Peak is due to a cancellation of effects, one of which would tend to increase the rate at higher altitudes and the other to decrease it. What are those effects? (Hint: Use the ideal-gas equation of state in your solution, and think about how the oxygen concentration at a high altitude would likely affect your breathing rate.)

After being purged with nitrogen, a low-pressure tank used to store flammable liquids is at a total pressure of 0.03 psig. (a) If the purging process is done in the moming when the tank and its contents are at \(55^{\circ} \mathrm{F}\), what will be the pressure in the tank when it is at \(85^{\circ} \mathrm{F}\) in the afternoon? (b) If the maximum design gauge pressure of the tank is 8 inches of water, has the design pressure been exceeded? (c) Speculate on the purpose of purging the tank with nitrogen.
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