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Chem 301 Problem Day # Fall 2008 Answers
- Derivatives: For each of the following find the first and second derivatives of the given function.
a. f(x)=x^3 +3x^2 +
3 6 2 6 6
2 (^2) x dx
d f x x dx
df
b. f(θ)=sin^2 θ
2 2 2
2 2 sin sin 2 sin cos 2 sin sin cos cos 2 sin cos dx
d f d
d dx
df
c. f(x)=ln(1+x) (first derivative only)
dx x
df 1
- Integrals: Do each of the indicated indefinite or definite integrals.
a. dx x^2
x
b. dx x
(^33) ln( x ) x
c.
2
1
( x^3 3 x^21 ) dx
3
(^24)
1
3
4 x x x
d. e^4 xdx 2
(^3) x x e^4 e^4 8
- For the function: f(x,y)=y ln(x^2 ) + x/y
y
f x x y
y y
x x
y
2 x
f y^2
ln( 2 ) y
x x
Other problems (pay attention to units!):
- On a hot day (40°C, 104°F), a He balloon has an internal pressure of 140 kPa and a volume of 2.46L. How many moles of gas does the balloon contain?
T = 40+273 = 313 K PV = nRT V = 2.46 dm^3 n = PV/RT = 1.40(2.40) / (.083145)(313) P = 140 kPa = 1.40 bar n = 0.132 moles
- A vessel contains 628 torr of Br 2 and 1.43 atm of CH 4 at 70ºC. If the temperature of the container falls to 14ºC, what is the new total pressure of the container?
PBr2 = 628 torr x 133.322 Pa/torr x bar / 100000 Pa = 0.837 bar PCH4 = 1.43 atm x 1.01325 bar/1atm = 1.45 bar Ptotal = 0.837 + 1.45 = 2.29 bar P 1 /T 1 = nR/V = P 2 /T 2 T 1 = 70 + 273 = 343 K 2.29/343 = P 2 / T 2 = 14 + 273 = 287 K P 2 = 1.91 bar
- A manufacturing company is selling stainless steel cylinders that contain 11.6 kg of ethylene (C 2 H 4 ) to a new processing plant in Detroit. When the tanks left the factory, it was a beautiful day (23°C) and the internal pressure of the tanks was 102 atm. When the tanks arrived in Detroit, it was still a beautiful day (23°C) and it was discovered that the tanks were not sealed properly. The new pressure in the tanks was only 74.8 atm. What is the mass of the ethylene that remains in the tanks?
T 1 = 23 + 273 = 296 V 1 = n 1 RT 1 /P 1 = (414)(.083145)(296)/(103.4) P 1 = 102 atm = 103.4 bar = 98.7 L n 1 = 11600 g x 1 mole/28.06 g = 414 mole
T 2 = 296 n 2 = P 2 V 2 /RT 2 = (75.8)(98.7)/(.083145)(296) P 2 = 74.8 atm = 75.8 bar = 304 mole V 2 = 98.7 L mass = 304 mole x 28.06 g /mole = 8.51 kg n 2 = ??
- Several ‘basic’ SI units (sec., kg, and meters) are the basis for several fundamental units in P- chem. a. Express the Pascal unit in terms of seconds, kg, and meters. According to Table 1.1, the Pascal is a N/m^2 which equates to kg·m-1·sec-
b. Energy will be a focus of much of our work this semester. The SI units for energy are Joules (J). Express the Joule unit in terms of seconds, kg, and meters. Energy is the work done by a force along a fixed distance: N·m. This is equivalent to kg·m^2 ·sec-
