# Equations of State for Ideal gas – Ideal gas law

** Phenomenological
thermodynamics is based on experiment, on measurements that you might make in a
lab, garage, or kitchen.

** For example, for any
fixed amount of a pure gas, two state variables are pressure, p, and volume, V.
Each can be controlled independently of each other.

** The pressure can be
varied while the volume is kept constant, or vice versa.

** Temperature, T, is
another state variable that can be changed independently from p and V.

** However, experience has
shown that if a certain pressure, volume, and temperature were specified for a
particular sample of gas at equilibrium, then all measurable, macroscopic properties
of that sample have certain specific values.

** That is, these three
state variables determine the complete state of our gas sample. Notice that we
are implying the existence of one other state variable: amount.

** The amount of material in
the system, designated by n, is usually given in units of moles.

** Further, arbitrary values
for all four variables p, V, n, and T are not possible simultaneously. Again,
experience (that is, experiment) shows this.

** It turns out that only
two of the three state variables p, V, and T are truly independent for any
given amount of a gas. Once two values are specified, then the third one must
have a certain value. This means that there is a mathematical equation into
which we can substitute for two of the variables and calculate what the
remaining variable must be.

** Say such an equation
requires that we know p and V and lets us calculate T. Mathematically, there
exists some function F such that:

where the function is
written as F(p, V) to emphasize that the variables are pressure and volume, and
that the outcome yields the value of the temperature T.

** Equations like equation
1.1 are called equations of state.

** One can also define
equations of state that yield p or V instead of T. In fact, many equations of
state can be algebraically rearranged to yield one of several possible state
variables.

####
**Ideal gas law**

** The earliest equations of
state for gases were determined by Boyle, Charles, Amontons, Avogadro,
Gay-Lussac, and others.We know these equations as the various gas laws.

**(1)**

**In the case of**

*, the equation of state involves multiplying the pressure by the volume to get a number whose value depended on the temperature of the gas:*

**Boyle’s gas law****(2)**whereas

*involves volume and temperature:*

**Charles’s gas law****(3)**

*relates volume and amount, but at fixed temperature and pressure:*

**Avogadro’s law**
** In the above three
equations, if the temperature, pressure, or amount were kept constant, then the
respective functions F(T), F(p), and F(n) would be constants. This means that
if one of the state variables that can change does, the other must also change
in order for the gas law to yield the same constant. This leads to the familiar
predictive ability of the above gas laws using the forms

** Similarly, using
equations 1.3 and 1.4, we can get:

** All three gas laws
involve volume, and they can be rewritten as:

** where the symbol α means
“is proportional to.’’We can combine the three proportionalities above into
one:

** Since p, V, T, and n are
the only four independent state variables for a gas, the proportionality form
of equation 1.8 can be turned into an equality by using a proportionality
constant:

** where we use R to
represent the proportionality constant. This equation of state relates the
static (unchanging) values of p, V, T, and n, not changes in these values. It
is usually rewritten as:

** which is the familiar
ideal gas law, with R being the ideal gas law constant.

####
**Fahrenheit and Celsius temperature**

** At this point, we must
return to a discussion of temperature units and introduce the proper
thermodynamic temperature scale.

** It has already been
mentioned that the Fahrenheit and Celsius temperature scales have arbitrary
zero points.

** What is needed is a temperature
scale that has an absolute zero point that is physically relevant.Values for
temperature can then be scaled from that point.

** In 1848, the British
scientist William Thomson, later made a baron and taking the title Lord Kelvin,
considered the temperature-volume relationship of gases and other concerns
(some of which we will address in future chapters) and proposed an absolute
temperature scale where the minimum possible temperature is about -273°C, or
273 Celsius-sized degrees below the freezing point of water. [A modern value is
-273.15°C, and is based on the triple point of H

_{2}O, not the freezing point.]. A scale was established by making the degree size for this absolute scale the same as the Celsius scale.
** In thermodynamics, gas
temperatures are almost always expressed in this new scale, called the absolute
scale or the Kelvin scale, and the letter K is used (without a degree sign) to
indicate a temperature in kelvins.

** Because the degree sizes
are the same, there is a simple conversion between a temperature in degrees
Celsius and the same temperature in kelvins:

** Occasionally, the conversion
is truncated to three significant figures and becomes simply K= °C + 273.

** In all of the gas laws
given above, the temperature must be expressed in kelvins! The absolute
temperature scale is the only appropriate scale for thermodynamic temperatures.
(For changes in temperature, the units can be kelvins or degrees Celsius, since
the change in temperature will be the same. However, the absolute value of the
temperature will be different.)

####
**The ideal gas law constant**

** Having established the
proper temperature scale for thermodynamics, we can return to the constant R.

** This value, the ideal gas
law constant, is probably the most important physical constant for macroscopic
systems.

** Specific numerical value
of R depends on the units used to express the pressure and volume, since the
units in an equation must also satisfy certain algebraic necessities.

** Table shows lists various
values of R.

####
**Notes**

** The ideal gas law is the
best known equation of state for a gaseous system.

** Gas systems whose state
variables p, V, n, and T vary according to the ideal gas law satisfy one
criterion of an ideal gas .

** Real gases, which do not
follow the ideal gas law exactly, can approximate ideal gases if they are kept
at high temperature and low pressure.

####
**Standard temperature and pressure (STP)**

** It is useful to define a
set of reference state variables for gases, since they can have a wide range of
values that can in turn affect other state variables.

** The most common set of
reference state variables for pressure and temperature is p = 1.0 atm and T =
273.15 K = 0.0°C.

** These conditions are
called standard temperature and pressure, abbreviated STP. Much of the
thermodynamic data reported for gases are given for conditions of STP.

** SI also defines standard
ambient temperature and pressure, SATP, as 273.15 K for temperature and 1 bar
for pressure (1 bar = 0.987 atm).

**Example: Calculate the volume of 1 mole of an ideal gas at SATP.**

**Solution:**

Using the ideal gas law and
the appropriate value for R:

This is slightly larger than
the commonly used molar volume of a gas at STP (about 22.4 L), since the
pressure is slightly lower.

**Reference:***physical Chemistry /David W. Ball / Cleveland State University /2011 .*

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