13.3 The ideal gas law  (Page 2/11)

 Page 2 / 11

At room temperatures, collisions between atoms and molecules can be ignored. In this case, the gas is called an ideal gas, in which case the relationship between the pressure, volume, and temperature is given by the equation of state called the ideal gas law.

Ideal gas law

The ideal gas law    states that

$\text{PV}=\text{NkT},$

where $P$ is the absolute pressure of a gas, $V$ is the volume it occupies, $N$ is the number of atoms and molecules in the gas, and $T$ is its absolute temperature. The constant $k$ is called the Boltzmann constant    in honor of Austrian physicist Ludwig Boltzmann (1844–1906) and has the value

$k=1\text{.}\text{38}×{\text{10}}^{-\text{23}}\phantom{\rule{0.25em}{0ex}}\text{J}/\text{K}.$

The ideal gas law can be derived from basic principles, but was originally deduced from experimental measurements of Charles’ law (that volume occupied by a gas is proportional to temperature at a fixed pressure) and from Boyle’s law (that for a fixed temperature, the product $\text{PV}$ is a constant). In the ideal gas model, the volume occupied by its atoms and molecules is a negligible fraction of $V$ . The ideal gas law describes the behavior of real gases under most conditions. (Note, for example, that $N$ is the total number of atoms and molecules, independent of the type of gas.)

Let us see how the ideal gas law is consistent with the behavior of filling the tire when it is pumped slowly and the temperature is constant. At first, the pressure $P$ is essentially equal to atmospheric pressure, and the volume $V$ increases in direct proportion to the number of atoms and molecules $N$ put into the tire. Once the volume of the tire is constant, the equation $\text{PV}=\text{NkT}$ predicts that the pressure should increase in proportion to the number N of atoms and molecules .

Calculating pressure changes due to temperature changes: tire pressure

Suppose your bicycle tire is fully inflated, with an absolute pressure of $7\text{.}\text{00}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{Pa}$ (a gauge pressure of just under $\text{90}\text{.}0\phantom{\rule{0.25em}{0ex}}{\text{lb/in}}^{2}$ ) at a temperature of $\text{18}\text{.}0\text{º}\text{C}$ . What is the pressure after its temperature has risen to $\text{35}\text{.}0\text{º}\text{C}$ ? Assume that there are no appreciable leaks or changes in volume.

Strategy

The pressure in the tire is changing only because of changes in temperature. First we need to identify what we know and what we want to know, and then identify an equation to solve for the unknown.

We know the initial pressure ${P}_{0}=7\text{.00}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{Pa}$ , the initial temperature ${T}_{0}=\text{18}\text{.}0ºC$ , and the final temperature ${T}_{\text{f}}=35\text{.}0ºC$ . We must find the final pressure ${P}_{\text{f}}$ . How can we use the equation $\text{PV}=\text{NkT}$ ? At first, it may seem that not enough information is given, because the volume $V$ and number of atoms $N$ are not specified. What we can do is use the equation twice: ${P}_{0}{V}_{0}={\text{NkT}}_{0}$ and ${P}_{\text{f}}{V}_{\text{f}}={\text{NkT}}_{\text{f}}$ . If we divide ${P}_{\text{f}}{V}_{\text{f}}$ by ${P}_{0}{V}_{0}$ we can come up with an equation that allows us to solve for ${P}_{\text{f}}$ .

$\frac{{P}_{\text{f}}{V}_{\text{f}}}{{P}_{0}{V}_{0}}=\frac{{N}_{\text{f}}{\text{kT}}_{\text{f}}}{{N}_{0}{\text{kT}}_{0}}$

Since the volume is constant, ${V}_{\text{f}}$ and ${V}_{0}$ are the same and they cancel out. The same is true for ${N}_{\text{f}}$ and ${N}_{0}$ , and $k$ , which is a constant. Therefore,

$\frac{{P}_{\text{f}}}{{P}_{0}}=\frac{{T}_{\text{f}}}{{T}_{0}}\text{.}$

We can then rearrange this to solve for ${P}_{\text{f}}$ :

${P}_{\text{f}}={P}_{0}\frac{{T}_{\text{f}}}{{T}_{0}},$

where the temperature must be in units of kelvins, because ${T}_{0}$ and ${T}_{\text{f}}$ are absolute temperatures.

Solution

1. Convert temperatures from Celsius to Kelvin.

$\begin{array}{}{T}_{0}=\left(\text{18}\text{.}0+\text{273}\right)\text{K}=\text{291 K}\\ {T}_{\text{f}}=\left(\text{35}\text{.}0+\text{273}\right)\text{K}=\text{308 K}\end{array}$

2. Substitute the known values into the equation.

${P}_{\text{f}}={P}_{0}\frac{{T}_{\text{f}}}{{T}_{0}}=7\text{.}\text{00}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{Pa}\left(\frac{\text{308 K}}{\text{291 K}}\right)=7\text{.}\text{41}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{Pa}$

Discussion

The final temperature is about 6% greater than the original temperature, so the final pressure is about 6% greater as well. Note that absolute pressure and absolute temperature must be used in the ideal gas law.

the meaning of phrase in physics
is the meaning of phrase in physics
Chovwe
write an expression for a plane progressive wave moving from left to right along x axis and having amplitude 0.02m, frequency of 650Hz and speed if 680ms-¹
how does a model differ from a theory
what is vector quantity
Vector quality have both direction and magnitude, such as Force, displacement, acceleration and etc.
Besmellah
Is the force attractive or repulsive between the hot and neutral lines hung from power poles? Why?
what's electromagnetic induction
electromagnetic induction is a process in which conductor is put in a particular position and magnetic field keeps varying.
Lukman
wow great
Salaudeen
what is mutual induction?
je
mutual induction can be define as the current flowing in one coil that induces a voltage in an adjacent coil.
Johnson
how to undergo polarization
show that a particle moving under the influence of an attractive force mu/y³ towards the axis x. show that if it be projected from the point (0,k) with the component velocities U and V parallel to the axis of x and y, it will not strike the axis of x unless u>v²k² and distance uk²/√u-vk as origin
show that a particle moving under the influence of an attractive force mu/y^3 towards the axis x. show that if it be projected from the point (0,k) with the component velocities U and V parallel to the axis of x and y, it will not strike the axis of x unless u>v^2k^2 and distance uk^2/√u-k as origin
No idea.... Are you even sure this question exist?
Mavis
I can't even understand the question
yes it was an assignment question "^"represent raise to power pls
Gabriel
Gabriel
An engineer builds two simple pendula. Both are suspended from small wires secured to the ceiling of a room. Each pendulum hovers 2 cm above the floor. Pendulum 1 has a bob with a mass of 10kg . Pendulum 2 has a bob with a mass of 100 kg . Describe how the motion of the pendula will differ if the bobs are both displaced by 12º .
no ideas
Augstine
if u at an angle of 12 degrees their period will be same so as their velocity, that means they both move simultaneously since both both hovers at same length meaning they have the same length
Modern cars are made of materials that make them collapsible upon collision. Explain using physics concept (Force and impulse), how these car designs help with the safety of passengers.
calculate the force due to surface tension required to support a column liquid in a capillary tube 5mm. If the capillary tube is dipped into a beaker of water
find the time required for a train Half a Kilometre long to cross a bridge almost kilometre long racing at 100km/h
method of polarization
Ajayi
What is atomic number?
The number of protons in the nucleus of an atom
Deborah
type of thermodynamics
oxygen gas contained in a ccylinder of volume has a temp of 300k and pressure 2.5×10Nm