w =
ð
f x
ð Þdx =
ð
f ext l o r
ð Þd l o r
ð Þ = l o
ð r 2
r 1
f ext r
ð Þdr
(2.17)
Thus, as long as we can measure the integrated force between r 1 and r 2 ,
we can determine the work done in stretching the DNA strand from r 1 to
r 2 (see Problem 1).
At this point it is worth noting that work done depends on the path taken
from the initial to the final state. We have clearly seen that the values of w
are different depending on whether a process is carried out in a single step
(irreversible) or very slowly (reversible). We say that work is a path
function: its value depends on the path taken. This is in contrast to other
functions that only depend on the difference between the final and initial
state, such as energy. These are known as state functions.
Finally, it is worth noting that we can write the general inequality w rev <
w irr , for any process. The inequality holds for expansion or compression,
being mindful of the sign convention used. For example, in the expansion
case, the system is doing work (-w), indicating maximum work for the
reversible process, while the converse is true for compression.
V
Piezo translator
x
Optical bar bends
when pulled
Force
(a)
(b)
200
160
120
80
40
0
Force (pN)
0.6 0.8 1.0 1.2 1.4 1.6
r
(i)
(ii)
(iv)
(iii)
Figure 2.10 (a) The stretching of a DNA strand using a
piezoelectric translator. (b) A
plot of the force required to
stretch the strand versus length
of the strand. A detailed experimental on the use of optical
tweezers to stretch DNA can be
found in Smith, S. B., Cui, Y., and
Bustamante, C. Science New
Series, Feb. 9, 1996, 271(5250):
795–799.
THE FIRST LAW OF THERMODYNAMICS
35