Transcript Heat Capacity II

CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2009
Heat Capacity II
Sound Velocity Method
Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2009
Heat Capacity II
Sound Velocity Method
Sound Velocity vS can be expressed as:
vS 
Where:

CP

CV
 
RT
1
2
M
Heat capacity @ constant pressure
Heat capacity @ constant volume
R – universal gas constant
T – temperature
M – molecular weight
If we know the speed of sound, we can calculate 
Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2009
Heat Capacity II
Sound Velocity Method
What the sound is?
Definition of sound:
Vibrations transmitted through an elastic solid or a liquid or gas,
with frequencies in the approximate range of 20 Hz to 20 kHz
Characteristics of sound:
f – frequency [number of vibrations per unit of time, Hz, or s-1]
 – wavelength [m]
By definition
vS

f
Speed of sound can be found as
v S  f
So, our goal is to measure the wavelength and the frequency of the sound
Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2009
Heat Capacity II
Sound Velocity Method
Experimental Setup
Scope
Generator
Y1
Y2
t
Microphone
Speaker

If the Generator output is connected to Y1 input of the Scope and
the Microphone output is connected to Y2 input, the Scope
shows 2 similar sin waveforms.
Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2009
Heat Capacity II
Sound Velocity Method
Experimental Setup
Scope
Generator
Y
y  y (x )
X
Microphone
Speaker

If the Generator output is connected to Y1 input of the Scope and
the Microphone output is connected to X input, the Scope shows
so called Lissajous pattern .
A Lissajous pattern is a graph of one frequency plotted on the y axis combined
with a second frequency plotted on the x axis. In our case both f are the same
Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2009
Heat Capacity II
Sound Velocity Method
Mathematical description of the simplest Lissajous pattern
Scope shows y as a function of x
y  y (x )
We know the time dependence both y and x
y (t )  Y0 sin( ft )
x(t )  X 0 sin( ft   )
Where  is the phase difference
between y(t) and x(t)
x(t) y(t)

Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2009
Heat Capacity II
Sound Velocity Method
Mathematical description of the simplest Lissajous pattern
y (t )  Y0 sin( ft )
x(t )  X 0 sin( ft   )
If  = 0
If  = p/2
(a quarter
of period)
y(x)
Assuming X0=Y0
y (t )  sin( ft )
x(t )  sin( ft )
x
y(x)=x
y (t )  sin( ft )
x(t )  sin( ft  p )  cos( ft )
2
y(x)
y 2 (t )  x 2 (t )  sin 2 ( ft)  cos 2 ( ft)  1
x
y2(t)+x2(t)=1
Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2009
Heat Capacity II
Sound Velocity Method
Mathematical description of the simplest Lissajous pattern
y (t )  Y0 sin( ft )
x(t )  X 0 sin( ft   )
If  = p
(a half
of period)
If  = 3p/2
(three quarters
of period)
y(x)
Assuming X0=Y0
y(x)=-x
y (t )  sin( ft )
x(t )  sin( ft  p )   sin( ft )
x
y (t )  sin( ft )
x(t )  sin( ft  3p )   cos( ft )
2
y(x)
y 2 (t )  x 2 (t )  sin 2 ( ft)  cos 2 ( ft)  1
x
y2(t)+x2(t)=1
Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2009
Heat Capacity II
Sound Velocity Method
Mathematical description of the simplest Lissajous pattern
y (t )  Y0 sin( ft )
x(t )  X 0 sin( ft   )
y (t )  sin( ft )
x(t )  sin( ft )
If  = 2p
(full period)
y(x)
Assuming X0=Y0
y(x)=x
x
Full period corresponds to 
Phase
difference
0
a quarter
of period,
or /4
a half
of period,
or /2
three quarters
of period,
or 3/4
full period,
or 
Figure on
the scope
Note: if X0 = Y0 the angle will be not 450 and the circles become the ellipses
Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2009
Heat Capacity II
Sound Velocity Method
How to measure the wavelength?
Frequency is given
by generator
Scope
Generator
Y
Microphone
Speaker
X

Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2009
Heat Capacity II
Sound Velocity Method
How to measure the wavelength?
Frequency is given
by generator
Scope
Generator
Y
Microphone
Speaker


X
4
Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2009
Heat Capacity II
Sound Velocity Method
How to measure the wavelength?
Frequency is given
by generator
Scope
Generator
Y
Microphone
Speaker


X
2
Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2009
Heat Capacity II
Sound Velocity Method
How to measure the wavelength?
Frequency is given
by generator
Scope
Generator
Y
Microphone
Speaker

3
X
4
Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2009
Heat Capacity II
Sound Velocity Method
How to measure the wavelength?
Frequency is given
by generator
Scope
Generator
Y
Microphone
Speaker

X

Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2009
Heat Capacity II
Sound Velocity Method
Summary
We measure the shape of Lissajous pattern as a function of
microphone position
Distance between two positions corresponding to straight 450
line is equal to the wavelength of the sound in given gas at
given sound frequency
We calculate the speed of sound in given gas at given
frequency using formula
v S  f
Instructor: Dr. Aleksey I. Filin