Ch 202 problem 3.69

A manufacturer of bicycles has 4815 wheels, 2305 frames, and 2255 handlebars.

(a) How many bicycles can be manufactured using these parts?

Asking how many bikes can be manufactured is the same as asking: what is the theoretical yield of bikes given 4815 wheels, 2305 frames, and 2255 handlebars?

we first need to know the relationship between wheels, frames, handlebars, and bicylcles. If we let wh = wheels, fr = frames, hb = handlebars, and bike = bicycle, then we can write a balanced 'equation' for making a bike:

2wh + 1fr + 1hb --> bike

let's set up some dummy units here:

While it seems pretty apparent that we'll get 2255 bikes, we need to be able to show this. Chemical stoichiometry problems aren't this obvious, so the ideas here are important. We'll use the general method discussed in class to find the limiting reactant: here, we'll convert frames to wheels and handlebars, and see if we have enough wheels and bars to use up all of the frames. We'll also convert bars to frames and wheels, and wheels to frames and bars, and see what it all means....

first, convert frames to wheels:

we need 4610 wheels to use up 2305 frames. We have 4815 wheels, so we're good.

next: bars to wheels

we need 4510 wheels to use up 2255 bars. Again, we have 4815 wheels, so we're OK.

wheels to bars:

we need 2408 bars to use up 4815 wheels - again, we only have 2255 bars!

wheels to frames:

we need 2408 frames to consume 4815 wheels. we only have 2305 frames!!!!

OK. We have enough wheels to consume all of the frames and bars. We need 2408 bars to consume all the wheels, and we need 2408 frames to consume all of the wheels. We don't have enough of either - so which one is the limiting reactant? Let's convert frames to bars.....

frames to bars:

we need 2305 bars to use up 2305 frames. we only have 2225 bars!

and bars to frames.....

we need 2255 frames to consume 2255 bars - we have 2305 frames, so we're golden.

so we conclude that we have enough wheels and frames and the number of bikes is limited by the number of bars - so the bars are the limiting reactant!

we convert the limiting reactant to product:

so we get 2255 bikes!

(b) How many parts of each kind are left over?

since bars are the limiting reactant, none are left......

to consume 2255 bars,

we start with 4815 wheels, and consume 4510, so

left

to consume 2305 frames,

we start with 2305 frames and use 2255, so

left

note - this problem was longer than usual because there were three 'reactants' and we had to do lots of converting. The usual case is two reactants, so we have fewer conversions to do and the results are easier to interpret!

DAC 7/3/08