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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