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Helium gas with a volume of \(3.20 \mathrm{~L}\), under a pressure of 0.180 atm and at \(41.0^{\circ} \mathrm{C}\), is warmed until both pressure and volume are doubled. (a) What is the final temperature? (b) How many grams of helium are there? The molar mass of helium is \(4.00 \mathrm{~g} / \mathrm{mol}\).

Short Answer

Expert verified
The final temperature is 355.24 degrees Celsius and the mass of helium is 0.088 grams.

Step by step solution

01

Calculate Initial Moles of Helium

To solve this problem, start by calculating the initial quantity of helium. The equation to use is \( PV = nRT \) where \( n \) is the quantity of molecules we aim to find out. Plugging in given values, \( P=0.180 \) atm, \( V=3.20 \) L, \( R=0.0821 \) L atm/mol K (known constant for the value of R), and the temperature \( T \), which needs to be converted from Celsius to Kelvin by the formula \( T(K)=T(C)+273.15 \), that is \( T=41.0+273.15=314.15 \) K. Solve for \( n \) to get \( n = \frac{PV}{RT} = \frac{0.180 * 3.20}{0.0821 * 314.15} = 0.022 \) moles.
02

Calculate Final Temperature

Since it is said that both pressure and volume of helium gas are doubled, the new values for pressure and volume are \( P'=2P=2 * 0.180 = 0.360 \) atm and \( V'=2V=2 * 3.20 = 6.40 \) L, respectively. Therefore, the final temperature \( T' \) can be found by inserting these new values and the previous result for moles \( n \) into the ideal gas law formula: \( P'V' = nRT' \) so \( T' = \frac{P'V'}{nR} = \frac{0.360 * 6.40}{0.022 * 0.0821} = 628.39 \) K, which is the final temperature.
03

Conversion of Final Temperature

Since the result was found in Kelvin, it can be converted back Celsius using the formula \( T(C)=T(K)-273.15 \), which gives us \( T'= 628.39-273.15= 355.24 \) C.
04

Calculate Mass of Helium

The mass \( m \) of the helium can be found by multiplying the previous result for moles \( n \) by the molar mass of helium, which is given as \( 4.00 \) g/mol. This gives us \( m=\(n * molar~mass)=0.022 * 4.00=0.088 \)g.

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Key Concepts

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

Gas Laws
Gas laws are fundamental for understanding the behavior of gases under different conditions of pressure, volume, and temperature. One of the most well-known gas laws is the Ideal Gas Law, expressed as \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the amount of substance in moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.
The Ideal Gas Law assumes that the gas consists of many randomly moving particles that obey the basic assumptions of kinetic theory. Despite its name, the ideal gas law also applies to real gases under high temperature and low pressure conditions.
In the given exercise, we saw the application of the Ideal Gas Law to solve for the number of moles of helium initially present, and subsequently to find out the final temperature when the pressure and volume were both doubled. Such problems reinforce the understanding that, by manipulating one or more of these variables, we can solve for unknowns in a predictable manner.
Temperature Conversion
Temperature conversion is an essential tool when working with gases, as many gas law equations require temperatures to be in Kelvin. The Kelvin scale is based on absolute zero, the point at which there is no movement in gas particles, making it perfect for absolute thermodynamic measurements.
To convert Celsius to Kelvin, the equation is simple: add 273.15 to the Celsius temperature. For example, converting 41.0°C to Kelvin results in: \( T(K)=41.0 + 273.15 = 314.15 \text{ K} \). In contrast, if you need to convert back to Celsius, subtract 273.15 from the Kelvin temperature: \( T(C)=T(K)-273.15 \).
This conversion is crucial in gas law calculations, as it ensures that temperature differences and proportionalities remain consistent across physics and chemistry, facilitating accurate and reliable results.
Molar Mass Calculation
Calculating the molar mass of a substance allows us to convert between grams and moles, a necessity in many chemical calculations, including those involving gases. In gas law problems, knowing the molar mass of the gas in question is crucial for determining its mass when given the amount of substance in moles.
The molar mass can be found using the periodic table, where the atomic or molecular weight is provided in \( \text{g/mol} \). For helium, the molar mass is given to be 4.00 \( \text{g/mol} \). To calculate the mass of helium, the number of moles \( n \) is multiplied by the molar mass: \( m = n \times \text{molar mass} \).
In the exercise, we calculated the molar mass of helium needed to find that the gas had a mass of 0.088 grams, reinforcing the link between molar mass and its practical utility in determining the mass of substances in gas law equations.

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Most popular questions from this chapter

You blow up a spherical balloon to a diameter of \(50.0 \mathrm{~cm}\) until the absolute pressure inside is 1.25 atm and the temperature is \(22.0^{\circ} \mathrm{C}\). Assume that all the gas is \(\mathrm{N}_{2},\) of molar mass \(28.0 \mathrm{~g} / \mathrm{mol}\). (a) Find the mass of a single \(\mathrm{N}_{2}\), molecule. (b) How much translational kinetic energy does an average \(\mathrm{N}_{2}\) molecule have? (c) How many \(\mathrm{N}_{2}\) molecules are in this balloon? (d) What is the total translational kinetic energy of all the molecules in the balloon?

A hot-air balloon stays aloft because hot air at atmospheric pressure is less dense than cooler air at the same pressure. If the volume of the balloon is \(500.0 \mathrm{~m}^{3}\) and the surrounding air is at \(15.0^{\circ} \mathrm{C}\) what must the temperature of the air in the balloon be for it to lift a total load of \(290 \mathrm{~kg}\) (in addition to the mass of the hot air)? The density of air at \(15.0^{\circ} \mathrm{C}\) and atmospheric pressure is \(1.23 \mathrm{~kg} / \mathrm{m}^{3}\).

A Jaguar XK8 convertible has an eight-cylinder engine. At the beginning of its compression stroke, one of the cylinders contains \(499 \mathrm{~cm}^{3}\) of air at atmospheric pressure \(\left(1.01 \times 10^{5} \mathrm{~Pa}\right)\) and a temperature of \(27.0^{\circ} \mathrm{C}\). At the end of the stroke, the air has been compressed to a volume of \(46.2 \mathrm{~cm}^{3}\) and the gauge pressure has increased to \(2.72 \times 10^{6} \mathrm{~Pa}\). Compute the final temperature.

A person at rest inhales \(0.50 \mathrm{~L}\) of air with each breath at a pressure of 1.00 atm and a temperature of \(20.0^{\circ} \mathrm{C}\). The inhaled air is \(21.0 \%\) oxygen. (a) How many oxygen molecules does this person inhale with each breath? (b) Suppose this person is now resting at an elevation of \(2000 \mathrm{~m}\) but the temperature is still \(20.0^{\circ} \mathrm{C}\). Assuming that the oxygen percentage and volume per inhalation are the same as stated above, how many oxygen molecules does this person now inhale with each breath? (c) Given that the body still requires the same number of oxygen molecules per second as at sea level to maintain its functions, explain why some people report "shortness of breath" at high elevations.

Three moles of an ideal gas are in a rigid cubical box with sides of length \(0.300 \mathrm{~m}\). (a) What is the force that the gas exerts on each of the six sides of the box when the gas temperature is \(20.0^{\circ} \mathrm{C} ?\) (b) What is the force when the temperature of the gas is increased to \(100.0^{\circ} \mathrm{C} ?\)

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