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Chapter 3: Problem 33

What is the internal energy of 6.00 mol of an ideal monatomic gas at \(200^{\circ} \mathrm{C}\) ?

Short Answer

Expert verified
The internal energy of the 6.00 mol of ideal monatomic gas at 200掳C is approximately 68.7475 kJ.

Step by step solution

01

Convert temperature to Kelvin

To convert the given temperature from Celsius to Kelvin, add 273.15 to the Celsius value: Temperature in Kelvin (T) = 200 + 273.15 = 473.15 K
02

Determine the number of moles and gas constant

In the problem, we are given n = 6.00 mol and R = 8.314 J/mol*K.
03

Calculate the internal energy

Using the formula for the internal energy of an ideal monatomic gas, we can now calculate the internal energy (E): E = (3/2) * n * R * T E = (3/2) * (6.00 mol) * (8.314 J/mol*K) * (473.15 K) Now, we can multiply the numbers to get the internal energy: E = 68.7475 kJ (approximately) The internal energy of the 6.00 mol of ideal monatomic gas at 200掳C is approximately 68.7475 kJ.

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Key Concepts

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

Ideal Monatomic Gas
An ideal monatomic gas is a theoretical concept often used in physics and chemistry. It is described as a gas where the particles move independently and are not influenced by intermolecular forces. This makes calculations much simpler and is often used as an approximation for real gases at high temperatures and low pressures.

  • "Monatomic" signifies that each particle in the gas is a single atom. Common examples include noble gases like helium (He) and neon (Ne).
  • The particles are assumed to be point masses, meaning they have mass but occupy no space.
  • The interactions are restricted to elastic collisions with each other or with the walls of the container.
This model is useful because it allows us to predict the behavior of gases with simple equations. In thermodynamics, it's used to derive various parameters, such as internal energy and pressure.
Temperature Conversion
Temperature is a crucial variable in thermodynamic calculations. It determines the energy of the particle movement in a gas. Celsius and Kelvin are two scales used for measuring temperature, and conversions between them are common in scientific contexts.

  • The Kelvin scale is the SI unit for temperature and starts at absolute zero, the theoretical lowest temperature possible.
  • Conversion from Celsius to Kelvin is straightforward: simply add 273.15 to the Celsius temperature:
The formula is:\[T(K) = T(掳C) + 273.15\]This conversion is crucial for using formulas like the one for the internal energy of gases, which requires temperature in Kelvin to ensure consistency with other units.
Moles of Gas
The mole is a fundamental unit in chemistry used to measure any substance. It counts atoms, molecules, or other entities and is defined as exactly \(6.022 \ imes 10^{23}\) entities per mole, known as Avogadro's number.

Understanding moles is central to stoichiometry, which involves calculating reactants and products in chemical reactions.
  • In gas calculations, moles help relate volume, pressure, temperature, and internal energy using the ideal gas law and related formulas.
  • Knowing the number of moles allows you to determine the amount of substance participating in a reaction or exerting pressure in a confined space.
In our problem, 6.00 moles of gas are used to calculate the internal energy, tying directly into the gas's thermodynamic properties.
Gas Constant
The gas constant, denoted as \(R\), is a fundamental constant in chemistry and physics. It appears in various equations concerning gases, linking physical quantities like pressure, volume, and temperature.

  • The value of \(R\) is constant, and for ideal gases, it's typically given as \(8.314 \, \text{J/mol鈰匥}\).
  • The gas constant is derived from the ideal gas law formula: \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin.
  • In thermodynamics, \(R\) enables calculations of energy and other properties of gases under assumed ideal conditions.
In the calculation of the internal energy of gases, it's used alongside the number of moles and temperature, illustrating its key role in these processes.

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Most popular questions from this chapter

When a gas expands isothermally, it does work. What is the source of energy needed to do this work?

In an adiabatic process, oxygen gas in a container is compressed along a path that can be described by the following pressure in atm as a function of volume \(\mathrm{V}\), with \(V_{0}=1 L: p=(3.0 \mathrm{atm})\left(V / V_{0}\right)^{-1.2} .\) The initial and final volumes during the process were 2 L and 1.5 L, respectively. Find the amount of work done on the gas.

Four moles of a monatomic ideal gas in a cylinder at \(27^{\circ} \mathrm{C}\) is expanded at constant pressure equal to \(1 \mathrm{atm}\) until its volume doubles. (a) What is the change in internal energy? (b) How much work was done by the gas in the process? (c) How much heat was transferred to the gas?

The state of 30 moles of steam in a cylinder is changed in a cyclic manner from a-b-c-a, where the pressure and volume of the states are: a \((30 \mathrm{atm}, 20 \mathrm{L}), \mathrm{b}(50 \mathrm{atm}, 20\) L), and \(c(50 \text { atm, } 45\) L). Assume each change takes place along the line connecting the initial and final states in the pV plane. (a) Display the cycle in the pV plane. (b) Find the net work done by the steam in one cycle. (c) Find the net amount of heat flow in the steam over the course of one cycle.

A cylinder containing three moles of nitrogen gas is heated at a constant pressure of 2 atm. The temperature of the gas changes from \(300 \mathrm{K}\) to \(350 \mathrm{K}\) as a result of the expansion. Find work done (a) on the gas, and (b) by the gas by using van der Waals equation of state instead of ideal gas law.
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