Background Background
The ideal gas law is given The ideal gas law is given byby
(1)(1)Where
Where p p is the absolute pressure,is the absolute pressure, V V is the volume, is the volume, nn is the number of moles, is the number of moles, R R is the universal is the universal gas constant and
gas constant and T T is the absolute temperature. Although this equation is widely by engineer and is the absolute temperature. Although this equation is widely by engineer and scientist, it is accurate over only a limited range of
scientist, it is accurate over only a limited range of pressure and temperature.pressure and temperature.
Furthermore, Eqs (1) is more appropriate for some gases that for others. An alternative equation Furthermore, Eqs (1) is more appropriate for some gases that for others. An alternative equation of state for gasses is given by:
of state for gasses is given by:
(( ))
(2)(2)Known as the
Known as the van der Waals equationvan der Waals equation, where, where v=V/nv=V/n is the molal volume and is the molal volume and aa and and bb are are empirical constants that depend on the
empirical constants that depend on the particular gas.particular gas.
The Task The Task
An engineering design project requires that you accurately estimate the molal volume ( An engineering design project requires that you accurately estimate the molal volume (nn)of)of acetone (
acetone (aa= 14.09 and= 14.09 and bb = 0.00994) for a number of different temperature from 300k, 400k and = 0.00994) for a number of different temperature from 300k, 400k and 500K and
500K and p p of 2.5 atm so that appropriate containment vessels can be selected. It is also of of 2.5 atm so that appropriate containment vessels can be selected. It is also of interest to examine how well acetone conforms to the ideal gas law by comparing the molal interest to examine how well acetone conforms to the ideal gas law by comparing the molal volume as calculated by Eqs (1) and (2).
volume as calculated by Eqs (1) and (2).
Instruction Instruction
1.
1. By using any of the numerical roots of equations method, solve the task above andBy using any of the numerical roots of equations method, solve the task above and determine the roots for
determine the roots for val der Waals equationsval der Waals equations using acetone gas using acetone gas
(CO2/PO1/C3) (CO2/PO1/C3) 2.
2. Show justification for your choice of method to solve Show justification for your choice of method to solve task above.task above.
(CO2/PO1/C3) (CO2/PO1/C3) GIVEN : R = 0.082054
Roots of equation is one of the numerical methods which widely used in engineering field as in engineering, we rarely come across a simple and straight forward situations which can be solve analytically. The problems are usually very complex and difficult to solve. Therefore, alternative approach provide by numerical method can be used to solve by approximating the true solution using computational implementation.
Roots of equation can be classified into two fundamental approaches : 1. Bracketing Method Bisection False position 2. Open method Fixed-point iteration Newton Rhapson Secant method Roots of polynomials
Bracketing methods start with two intial guesses that bracket the true roots while Open Method need no initial guesses to bracket the true root. Open method usually more efficient compared with bracketing method but not always work.
CALCULATION
From the question, the ideal gas law is :
(1)Where
p : absolute pressure V : volume
n : the number of moles R : the universal gas constant
T : the absolute temperature
From equation (1), the molal volume (v) can be determined:
The given values are :
Therefore, the molal volume (v) for each temperatures are :
Pressure, p (atm) Gas constant, R Temperature (K) Molal volume, v(L/mol)
2.5 0.082054 300 9.846480
2.5 0.082054 400 13.128640
2.5 0.082054 500 16.410800
An alternative equation of state for gases which is known as van der Waals is given by :
( )
(2)Where :
v =
: molal volume.a and b : empirical constants that depend on the particular gas.
From equation (2), we can simplified the equation into :
()
()
()
()
(3) By differentiate the simplified equation above, we obtained :()
()
(4)To solve the task given, the method of numerical equation chosen is Newton Rhapson where the first derivative is equivalent of the slope given b y:
(
(
)
)
F
irstly, we need to find the roots from equation (3) by substituting all the values given. The values given are :For T=300 K ,
()
From this equation, the roots obtained are
=0.010118882,
=9.247621045,
=0.598680072. Since
=9.247621045 is the nearest to the initial guess we obtained from equation (1), therefore
=9.247621045 is the true root for T=300 K. Initial guess,
= 9.846480By using acetone gas, the root for val der Waals equations at T=300K are9.247621.
i
+1 f(
) f’(
) εa(%) εt(%) 0 9.846480 9.314463 136.187600 255.983600 - 6.475816 1 9.314463 9.248599 13.551340 205.747900 0.712149 0.722803 2 9.248599 9.247621 0.195441 199.824000 0.010576 0.010578 3 9.247621 9.247621 0.000043 199.736500 0.000002 0.000002 4 9.247621 9.247621 0.000000 199.736500 0.000000 0.000000For T=400 K ,
()
From this equation, the roots obtained are
=12.69497233,
=0.010181469,
=0.433426199. Since
=12.69497233 is the nearest to the initial guess we obtained from equation (1), therefore
=12.69497233 is the true root for T=400 K. Initial guess,
= 13.128640By using acetone gas, the root for val der Waals equations at T=400K are12.694972.
i
+1 f(
) f’(
) εa(%) εt(%)0 13.128640 12.722287 180.559300 444.340500 3.41605841 1 12.722286 12.695091 10.667360 392.250400 0.214218944 0.21515739 2 12.695091 12.694972 0.046226 388.852700 0.000936417 0.00093644 3 12.694972 12.694972 0.000000 388.837900 0.000000 0.000000
For T=500 K ,
()
From this equation, the roots obtained are
=16.07024466,
=0.010245653,
=0.340247655. Since
=16.070244669 is the nearest to the initial guess we obtained from equation (1), therefore
=16.070244669 is the true root for T=500 K.Initial guess,
= 16.410800By using acetone gas, the root for val der Waals equations at T=500K are16.070247.
i
+1 f(
) f’(
) εa(%) εt(%)0 16.410800 16.083960 224.395700 686.560275 2.119154 1 16.083959 16.070270 8.675450 633.740575 0.085184 0.085331 2 16.070270 16.070247 0.014906 631.563254 0.000147 0.000147 3 16.070247 16.070247 0.000000 631.559551 0.000000 0.000000
DISCUSSION
The van der Waals equation is basically a complex adjustment to the ideal gas as the ideal gas law works only with ideal gases and it is not too accurate in real life situations. However, people choose to work with the ideal gas law rather than the van der Waals equation as in most cases, the difference between these two equation is as low as 1% or less. In this task, both equations are used to determine the roots of molal volume of acetone gas at three different temperatures (300K, 400K and 500K).
Temperature (K) Ideal Gas law van der Waals True roots
300 9.846480 9.247621
=0.010118882
=9.247621045
=0.598680072 400 13.128640 12.694972
=12.69497233
=0.010181469
=0.433426199 500 16.410800 16.070247
=16.07024466
=0.010245653
=0.340247655 The table above shows the values or roots obtained using ideal gas law, van der Waals and also the true roots. During the calculation using the Newton Rhapson method, the values obtained by ideal gas law are used as the initial guess. Comparing the molal volume obtained from both equation with the true roots, the values of root obtained using van der Waals are closer to the true root than the values obtained using ideal gas law.From the answers gained, there is slight difference of values obtained from ideal gas law and van der Waals equation. This is due to the properties of the formula that gives the answer a different value. For this task, a Newton Raphson method is used to determine the roots. The reason Newton Raphson is chosen is because this method is easily converged. When the initial
CONCLUSSION
Numerical methods are another approach that provides solution for many engineering problems as some cases cannot be solved using mathematical methods. There are few numerical methods and one of it is roots of equation methods which have been used to solve this task. There are two classes to determine the roots of equation which is bracketing method and open method. The chosen method for this task is Newton Rhapson which is from open method as it is an efficient method. To obtain the answers correctly, Microsoft Excel is used to solve this task. Other than Microsoft Excel, other software such as MATLAB can also be used.
REFERENCES
1. http://hyperphysics.phy-astr.gsu.edu/hbase/Kinetic/waal.html
2. MEC500 NUMERICAL METHOD CHAPTER 2 “Roots of Equations – Bracketing Method”
